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

Quantum spin liquids represent a novel phase of magnetic matter where quantum fluctuations are large enough to suppress the formation of local order parameters, even down to zero temperature. Quantum spin liquid states can emerge from frustrated quantum magnets. These states show several peculiar properties, such as topological order, fractional excitations, and long-range entanglement. The Kitaev spin model on the honeycomb lattice is one of the few models proposed which can exactly show the existence of a $\mathbb{Z}_2$ quantum spin liquid. The model describes spins featuring frustrated compass interactions, and it exhibits a quantum spin liquid ground state. The model's ground state can be found exactly by representing spins in terms of Majorana fermions. It turns out that spin excitations fractionalize into two degrees of freedom: spinless matter fermions and flux excitations of the emergent $\mathbb{Z}_2$ gauge theory. Recently, possible solid-state realizations of Kitaev quantum spin liquids have been proposed in a class of frustrated Mott insulators. Unfortunately, experiments can not unambiguously identify quantum spin liquids, due to their elusive nature. Nevertheless, indirect observations on a spin liquid state can be done by looking at its excitations. Along this line, thermal transport investigations provide for an option to study heat-carrying excitations, and thus the properties of the related spin liquid state. In this doctoral thesis work, I performed a study of longitudinal thermal transport properties in the two-dimensional Kitaev spin model. This study aims to advance the understanding of transport in prototypical frustrated quantum magnets that might harbor Kitaev physics, and in particular quantum spin liquid states. For this purpose, I explored the model for varying exchange coupling regimes $-$ to underline the impact of anisotropy on transport $-$ and I studied transport over a wide range of temperatures. Transport properties have been explored within the formalism of the linear response theory. Based on the latter, thermal transport coefficients can be evaluated by calculating dynamical energy-current auto-correlation functions. First, I performed an analytical study of the uniform gauge sector of the model $-$ where excitations of gauge degrees of freedom are neglected. Analytical findings for the energy-current correlations, and their related transport coefficients, imply a finite-temperature ballistic heat conductor in terms of free matter fermion excitations $-$ independent of exchange couplings. Second, thermal transport has been studied at finite temperatures, considering thermal gauge excitations off the uniform gauge sector. For this purpose, I made use of two complementary numerical methods able to treat finite-temperature systems. On the one hand, I resorted on the exact diagonalization of the Kitaev Hamiltonian given in terms of fermions and a real-space dependent $\mathbb{Z}_2$ gauge potential, to study relatively small systems. On the other hand, I used an approximate method based on a mean-field treatment of thermal gauge fluctuations. The method allowed to extend the study of thermal transport to systems with up to $\sim\mathcal{O}(10^4)$ spinful sites. It made possible the computation of correlation functions by reducing the exact trace over all gauge states to an average over dominant gauge states suited to a given temperature range. The reliability of the method has been checked by comparing to numerically exact thermodynamics of systems. Based on the thermodynamic analysis, the method has been restricted to a temperature range where the mean-field treatment of gauge fluctuations is acceptable. Within such temperature range, the method succeeded in well reproducing exact results. The prime advantage of this method is its capability to reveal important features in the energy-current correlation spectra, not captured by the exact diagonalization approach because of finite-size effects. I found that the energy-current correlation spectra, in the presence of thermal gauge excitations, show clear signatures of spin fractionalization. In particular, the low-energy part of spectra displays features arising from a temperature-dependent matter-fermion density relaxation off an emergent thermal gauge disorder. This static gauge disorder also leads to the appearance of a pseudogap in the zero-frequency limit, which closes in the thermodynamic limit. The extracted dc heat conductivity is consequently influenced by this interplay between matter fermions and gauge degrees of freedom. The anisotropy in the exchange couplings moves Kitaev systems through gapless and gapped phases of the matter fermion sector. Effects of anisotropy are visible in the dc conductivities which display a low-temperature dependence crossing over from power-law to exponentially activated behavior upon entering the gapped phase. Therefore, I found that in the thermodynamic limit, two-dimensional Kitaev systems feature dissipative transport, regardless of exchange couplings. This finding is in contrast to the ballistic transport found discarding gauge excitations in the uniform gauge sector, which underlines the relevance of gauge degrees of freedom in thermal transport properties of Kitaev systems.

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.

In this PhD thesis, we study the interplay between symmetry-breaking order and quantum-disordered phases in the milieu of frustrated quantum magnets, and further show how the excitation process of long-wavelength (semi-)classical modes in spin-orbit coupled antiferromagnets crucially depends on the nature and interactions of the underlying quantum quasiparticles. First, we focus on Kitaev's exactly solvable model for a Z2 spin liquid as a building block for constructing novel phases of matter, utilizing Majorana mean-field theory (MMFT) to map out phase diagrams and study occurring phases. In the Kitaev Kondo lattice, conduction electrons couple via a Kondo interaction to the local moments in the Kitaev model. We find at small Kondo couplings a fractionalized Fermi liquid (FL*) phase, a stable non-Fermi liquid where conventional electronic quasiparticles coexist with the deconfined excitations of the spin liquid. The transition between FL* and a conventional Fermi liquid is masked by an exotic (confining) superconducting phase which exhibits nematic triplet pairing, which we argue to be mediated by the Majorana fermions in the Kitaev spin liquid. We moreover study bilayer Kitaev models, where two Kitaev honeycomb spin liquids are coupled via an antiferromagnetic Heisenberg interaction. Varying interlayer coupling and Kitaev coupling anisotropy, we find both direct transitions from the spin liquid to a trivial dimer paramagnet as well as intermediate 'macrospin' phases, which can be studied by mappings to effective transverse-field Ising models. Further, we find a novel interlayer coherent pi-flux phase. Second, we consider the stuffed honeycomb Heisenberg antiferromagnet, where recent numerical studies suggest the coexistence of collinear Néel order and a correlated paramagnet, dubbed 'partial quantum disorder'. We elucidate the mechanism which drives the disorder in this model by perturbatively integrating out magnons to derive an effective model for the disordered sublattice. This effective model is close to a transition between two competing ground states, and we conjecture that strong fluctuations associated with this transition lead to disorder. Third, we study the generation of coherent low-energy magnons using ultrafast laser pulses in the spin-orbit coupled antiferromagnet Sr2IrO4, inspired by recent pump-probe experiments. While the relaxation dynamics of the system at long time scales can be well described semi-classically, the ultrafast excitation process is inherently non-classical. Using symmetry analysis to write down the most general coupling between electric field and spin operators, we subsequently integrate out high-energy spin fluctuations to derive induced effective fields which act to excite the low-energy magnon, constituting a generalized 'inverse Faraday effect'. Our theory reveals a tight relationship between induced fields and the two-magnon density of states.:1 Introduction 1.1 Frustrated antiferromagnets 1.2 Quantum spin liquids 1.3 Fractionalization and topological order 1.4 Spin-orbit coupling 1.5 Outline I Novel phases by building on Kitaev’s honeycomb model 2 Kitaev honeycomb spin liquid 2.1 Microscopic spin model and constants of motion 2.2 Majorana representation of spin algebra 2.3 Exact solution 2.3.1 Ground state 2.3.2 Correlations and dynamics 2.3.3 Thermodynamic properties 2.4 Z2 gauge structure 2.5 Toric code 2.6 Topological order 2.6.1 Superselection sectors and ground-state degeneracy 2.6.2 Topological entanglement entropy 2.6.3 Symmetry-enriched and symmetry-protected topological phases 3 Mean-field theory 3.1 Generalized spin representations 3.1.1 Parton constructions 3.1.2 SO(4) Majorana representation 3.2 Projective symmetry groups 3.3 Mean-field solution of the Kitaevmodel 3.4 Comparisonwithexactsolution 3.4.1 Spectral properties 3.4.2 Correlation functions 3.4.3 Thermodynamic properties 3.5 Generalized decoupling 3.6 Comparison to previous Abrikosov fermion mean-field theories of the Kitaev model 3.7 Discussion 4 Fractionalized Fermi liquids and exotic superconductivity in the Kitaev Kondo lattice 4.1 Metals with frustration 4.2 Local-moment formation and Kondo effect 4.2.1 Single Kondo impurity 4.2.2 Kondo lattices and heavy Fermi liquids 4.3 Fractionalized Fermi liquids 4.4 Construction of the Kitaev Kondo lattice 4.4.1 Hamiltonian 4.4.2 Symmetries 4.5 Mean-field decoupling of Kondo interaction 4.5.1 Solution of self-consistency conditions 4.6 Overview of mean-field phases 4.7 Fractionalized Fermi liquid 4.7.1 Results from mean-field theory 4.7.2 Perturbation theory beyond mean-field theory 4.8 Heavy Fermi liquid 4.9 Superconducting phases 4.9.1 Spontaneously broken U(1) phase rotation symmetry 4.9.2 Excitation spectrum and nematicity 4.9.3 Topological triviality 4.9.4 Group-theoretical classification 4.9.5 Pairing glue 4.10 Comparison with a subsequent study 4.11 Discussion and outlook 5 Bilayer Kitaev models 5.1 Model and stacking geometries 5.1.1 Hamiltonian 5.1.2 Symmetries and conserved quantities 5.2 Previous results 5.3 Mean-field decoupling and phase diagrams 5.3.1 AA stacking 5.3.2 AB stacking 5.3.3 σAC stacking 5.3.4 σ ̄AC stacking 5.4 Quantum phase transition in the AA stacking 5.4.1 Perturbative analysis 5.5 Phase transition in the σAC stacking 5.6 Macro-spin phases 5.6.1 KSL-MAC transition: Effective model for Kitaev dimers 5.6.2 DIM-MAC transition: Effective theory for triplon condensation 5.6.3 Macro-spin interactions and series expansion results 5.6.4 Antiferromagnet in the AB stacking 5.7 Stability of KSL and the interlayer-coherent π-flux phase 5.7.1 Perturbative stability of the Kitaev spin liquid 5.7.2 Spontaneous interlayer coherence near the isotropic point 5.8 Summary and discussion II Partial quantum disorder in the stuffed honeycomb lattice 6 Partial quantum disorder in the stuffed honeycomb lattice 6.1 Definition of the stuffed honeycomb Heisenberg antiferromagnet 6.2 Previous numerical results 6.3 Derivation of an effective model 6.3.1 Spin-wave theory for the honeycomb magnons 6.3.2 Magnon-central spin vertices 6.3.3 Perturbation theory 6.3.4 Instantaneous approximation 6.3.5 Truncation of couplings 6.3.6 Single-ion anisotropy 6.3.7 Discussion of most dominant interactions 6.4 Analysis of effective model 6.4.1 Classical ground states 6.4.2 Stability of classical ground states in linear spin-wave theory 6.4.3 Minimal model for incommensurate phase 6.4.4 Discussion of frustration mechanism in the effective model 6.5 Partial quantum disorder beyond the effectivemodel 6.5.1 Competition between PD and the (semi-)classical canted state 6.5.2 Topological aspects 6.5.3 Experimental signatures 6.6 Discussion 6.6.1 Directions for further numerical studies 6.6.2 Experimental prospects III Optical excitation of coherent magnons 7 Ultrafast optical excitation of magnons in Sr2IrO4 7.1 Pump-probe experiments 7.2 Previous approaches to the inverse Faraday effect and theory goals 7.3 Sr2IrO4 as a spin-orbit driven Mott insulator 7.4 Spin model for basal planes in Sr2IrO4 7.4.1 Symmetry analysis 7.4.2 Classical ground state and linear spin-wave theory 7.4.3 Mechanism for in-plane anisotropy 7.5 Pump-induced dynamics 7.5.1 Coupling to the electric field: Symmetry analysis 7.5.2 Keldysh path integral 7.5.3 Low-energy dynamics 7.5.4 Driven low-energy dynamics 7.6 Derivation of the induced fields 7.6.1 Perturbation theory 7.6.2 Evaluation of loop diagram 7.6.3 Analytical momentum integration in the continuum limit 7.6.4 Numerical evaluation of effective fields 7.7 Analysis of induced fields 7.7.1 Polarization and angular dependence 7.7.2 Two-magnon spectral features 7.8 Applications to experiment 7.8.1 Predictions for experiment 7.8.2 Magnetoelectrical couplings 7.9 Discussion and outlook 8 Conclusion and outlook 8.1 Summary 8.2 Outlook IV Appendices A Path integral methods B Spin-wave theory B.1 Holstein-Primakoff bosons B.2 Linear spin-wave theory B.2.1 Diagonalization via Bogoliubov transformation B.2.2 Applicability of linear approximation B.3 Magnon-magnon interactions B.3.1 Dyson's equation and 1/S consistency B.3.2 Self-energy from quartic interactions in collinear states on bipartite lattices C Details on the SO(4) Majorana mean-field theory C.1 SO(4) Matrix representation of SU(2) subalgebras C.2 Generalized SO(4) Majorana mean-field theory for a Heisenberg dimer (Chapter 3) C.3 Dimerization of SO(4) Majorana mean-field for the Kitaev model (Chapter3) C.4 Mean-field Hamiltonian in the Kitaev Kondo lattice (Chapter 4) C.5 Example solutions in the superconducting phase for symmetry analysis (Chapter4) D Linear spin-wave theory for macrospin phase in the bilayer Kitaev model (Chapter 5) D.1 Spin-wave Hamiltonian and Bogoliubov rotation D.2 Results and discussion E Extrapolation of the effective couplings for the staggered field h -> 0 (Chapter 6) E.1 xy interaction E.1.1 Leadingorder ~ S0 E.1.2 Subleadingorder ~ S^(−1) E.2 z-Ising interaction F Light-induced fields by analytical integration (Chapter 7) F.1 Method F.2 Results Bibliography

Doctor of Philosophy / Electrical and Computer Engineering / Sanjoy Das / Quantum circuit synthesis is one of the major areas of current research in the field of quantum computing. Analogous to its Boolean counterpart, the task involves constructing arbitrary quantum gates using only those available within a small set of universal gates that can be realized physically. However, unlike the latter, there are an infinite number of single qubit quantum gates, all of which constitute the special unitary group SU(2). Realizing any given single qubit gate using a given universal gate family is a complex task. Although gates can be synthesized to arbitrary degree of precision as long as the set of finite strings of the gate family is a dense subset of SU(2), it is desirable to accomplish the highest level of precision using only the minimum number of universal gates within the string approximation. Almost all algorithms that have been proposed for this purpose are based on the Solovay-Kitaev algorithm. The crux of the Solovay-Kitaev algorithm is the use of a procedure to decompose a given quantum gate into a pair of group commutators with the pair being synthesized separately. The Solovay-Kitaev algorithm involves group commutator decomposition in a recursive manner, with a direct approximation of a gate into a string of universal gates being performed only at the last level, i.e. in the leaf nodes of the search tree representing the execution of the Solovay-Kitaev algorithm. The main contribution of this research is in integrating conventional optimization procedures within the Solovay-Kitaev algorithm. Two specific directions of research have been studied. Firstly, optimization is incorporated within the group commutator decomposition, so that a more optimal pair of group commutators are obtained. As the degree of precision of the synthesized gate is explicitly minimized by means of this optimization procedure, the enhanced algorithm allows for more accurate quantum gates to be synthesized than what the original Solovay-Kitaev algorithm achieves. Simulation results with random gates indicate that the obtained accuracy is an order of magnitude better than before. Two versions of the new algorithm are examined, with the optimization in the first version being invoked only at the bottom level of Solovay-Kitaev algorithm and when carried out across all levels of the search tree in the next. Extensive simulations show that the second version yields better results despite equivalent computation times. Theoretical analysis of the proposed algorithm is able to provide a more formal, quantitative explanation underlying the experimentally observed phenomena. The other direction of investigation of this research involves formulating the group commutator decomposition in the form of bi-criteria optimization. This phase of research relaxed the equality constraint in the previous approach and with relaxation, a bi-criteria optimization is proposed. This optimization algorithm is new and has been devised primarily when the objective needs to be relaxed in different stages. This bi-criteria approach is able to provide comparably accurate synthesis as the previous approach.

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.

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.

Magnetic frustration is a phenomenon arising in spin systems when spin interactions cannot all be satisfied at the same time. A typical example of geometric frustration is a triangle with Ising-spins at its vertices and antiferromagnetic interaction. While we can easily anti-align two neighbouring spins, it is not possible for the third one to simultaneously anti-align with both of them. Another flavour of magnetic frustration is the so called exchange frustration, where different spin components interact in an Ising fashion on different bonds. Moreover, frustrated spin systems give rise to exotic states of matter, such as spin liquids, spin ices and nematic phases. As frustrated systems are rarely analytically solvable, numerical techniques are of the utmost importance in this framework. This dissertation is concerned with a specific class of models, namely one- and quasi-one-dimensional spin systems and studies their properties by making use of the density matrix renormalisation group technique. This method has been shown to be extremely powerful and reliable to study chain and ladder models. We consider examples of both geometric and exchange frustration. For the former, we take into consideration one of the prototypical examples of geometric frustration in one dimension: the J1-J2 model with ferromagnetic nearest-neighbour interaction J1<0 and antiferromagnetic next-nearest-neighbour interaction J2>0. Our results show the existence of a Haldane gap supported by a special AKLT-like valence bond solid state in a specific region of the coupling ratio. Furthermore, we consider the effect of dimerisation of the first-neighbour coupling. This dimerisation affects the critical point and the ground state underlying the spin gap. These models are of interest in the context of cuprate chain materials such as LiVCuO4, LiSbCuO4 and PbCuSO4(OH)2. Concerning exchange frustration, we consider the celebrated Kitaev-Heisenberg model: it is an extension of the exactly solvable Kitaev model with an additional Heisenberg interaction. The Kitaev-Heisenberg model is currently the minimal model for candidate Kitaev materials. The extended model is not analytically solvable and numerics are needed to study the properties of the system. While both the original Kitaev and the Kitaev-Heisenberg models live on a honeycomb lattice, we here perform systematic studies of the Kitaev-Heisenberg chain and of the two-legged ladder. While the chain cannot support a Kitaev spin liquid state, it shows nevertheless a rich phase diagram despite being a one-dimensional system. The long-range ordered states of the honeycomb can be understood in terms of coupled chains within the Kitaev-Heisenberg model. Following this reasoning, we turn our attention to the Kitaev-Heisenberg model on a two-legged ladder. Remarkably, the phase diagram of the ladder is extremely similar to that of the honeycomb model and the differences can be explained in terms of the different dimensionalities. In particular, the ladder exhibits a topologically non-trivial phase with no long-range order, i.e., a spin liquid. Finally, we investigate the low-lying excitations of the Kitaev-Heisenberg model for both the chain and the ladder geometry.

京都大学 / 0048 / 新制・課程博士 / 博士(理学) / 甲第19479号 / 理博第4139号 / 新制||理||1595(附属図書館) / 32515 / 京都大学大学院理学研究科物理学・宇宙物理学専攻 / (主査)准教授 戸塚 圭介, 教授 川上 則雄, 教授 石田 憲二 / 学位規則第4条第1項該当 / Doctor of Science / Kyoto University / DFAM

No description available.