In this overview I describe the simplest models for the quantum Hall and quantum spin Hall effects, and give some general indications as to the description of topological insulators. As a background to the theoretical models I will first trace the development leading up to the description of topological insulators . Then I will present Laughlin's original model for the quantum Hall effect and briefly discuss its limitations. After that I will describe the Kane and Mele model for the quantum spin Hall effect in graphene and discuss its relation to a general quantum spin Hall system. I will conclude by giving a conceptual description of topological insulators and mention some potential applications of such states.

By numerical methods, two models of anisotropic mixed quantum spin chains, consisting of spins of two different sizes, Sa = 1/2 and Sb = 1 as well as Sb = 3/2, are studied with respect to their critical properties at quantum phase transitions in a selected region of parameter space. The quantum spin chains are made up of basecells of four spins, according to the structure Sa − Sa − Sb − Sb. They are described by the XXZ Hamiltonian, that extends the quantum Heisenberg model by a variable anisotropic exchange interaction. As additional control parameter, an alternating exchange constant between nearest-neighbour spins is introduced. Insight gained by complementary application of exact diagonalization and quantum Monte Carlo simulations, as well as appropriate methods of analysis, is embedded in the broad existing knowledge on homogeneous quantum spin chains. In anisotropic homogeneous quantum spin chains, there exist phase boundaries with continuously varying critical exponents, the Gaussian critical lines, along which, in addition to standard scaling relations, further extended scaling relations hold. Reweighting methods, also applied to improved quantum Monte Carlo estimators, and finite-size scaling analysis of simulation data deliver a wealth of numerical results confirming the existence of a Gaussian critical line also in the mixed spin models considered. Extrapolation of exact data offers, apart from confirmation of simulation data, furthermore, insight into the conformal operator content of the model with Sb = 1. / Mittels numerischer Methoden werden zwei Modelle anisotroper gemischter Quantenspinketten, bestehend aus Spins zweier unterschiedlicher Größen, Sa = 1/2 und Sb = 1 sowie Sb = 3/2, hinsichtlich ihrer kritischen Eigenschaften an Quanten-Phasenübergängen in einem ausgewählten Parameterbereich untersucht. Die Quantenspinketten sind aus Basiszellen zu vier Spins, gemäß der Struktur Sa − Sa − Sb − Sb, aufgebaut. Sie werden durch den XXZ Hamiltonoperator beschrieben, der das isotrope Quanten-Heisenberg Modell um eine variable anistrope Austauschwechselwirkung erweitert. Als zusätzlicher Kontrollparameter wird eine alterniernde Kopplungskonstante zwischen unmittelbar benachbarten Spins eingeführt. Die durch komplementäre Anwendung exakter Diagonalisierung und Quanten-Monte-Carlo Simulationen, sowie entsprechender Analyseverfahren, gewonnenen Erkenntnisse werden in das umfangreiche existierende Wissen über homogene Quantenspinketten eingebettet. Im Speziellen treten in anisotropen homogenen Quantenspinketten Phasengrenzen mit kontinuierlich variierenden kritischen Exponenten auf, die Gaußschen kritischen Linien, auf denen neben den herkömmlichen auch erweiterte Skalenrelationen Gültigkeit besitzen. Umgewichtungsmethoden, speziell auch angewandt auf verbesserte Quanten-Monte-Carlo Schätzer, und Endlichkeitsskalenanalyse von Simulationsdaten liefern eine Fülle von numerischen Ergebnissen, die das Auftreten der Gaußschen kritischen Linie auch in den untersuchten gemischten Quantenspinketten bestätigen. Die Extrapolation exakter Daten bietet, neben der Bestätigung der Simulationsdaten, darüber hinaus Einblick in einen Teil des konformen Operatorinhalts des Modells mit Sb = 1.

Este trabalho apresenta as propriedades de transporte eletrônico de isolantes topológicos bidimensionais (TI) baseados em poços quânticos de HgTe/CdTe. Estas heteroestruturas, no regime de bandas invertido, contem um novo estado conhecido como isolante de spin Hall quântico (QSHI). Este estado apresenta um comportamento de isolante no corpo (bulk), mas exibe estados condutores sem lacunas nas bordas (edges), as quais podem ser verificadas em medidas de transporte. Medidas de resistência de quatro terminais foram observadas perto do valor quantizado em amostras mesoscópicas. No entanto, para amostras com mais de um m, a resistência pode ser muito maiores que h/2e2 devido à presença de defasagem de spin, não homogeneidade ou desordem na amostra. Esta tese aborda o problema da resistência não quantizado observada em amostras macroscópicas de dimensões maiores a algum mícron. Nós relatamos observação e investigação sistemática de transporte local e não local em poços quânticos de HgTe (8.0-8.3 nm) com estrutura de banda invertida correspondente à fase de isolante de spin Hall quântico. O dispositivo MCT1 consiste de três segmentos consecutivos de largura 4 m e de comprimentos diferentes (2 m, 8 m, 32 m), e sete sondas de tensão. O dispositivo MCT2 foi fabricado com um comprimento litográfico de 6 m e largura 5 m. Ambos dispositivos estão equipados com uma porta superior (top gate), que permite ajustar a densidade de portadores do dispositivo. A aplicação de uma tensão de porta muda a densidade de portadores, transformando a condutividade do poço quântico de tipo n para tipo p através de uma fase intermediária chamada de ponto a neutralidade de carga (CNP). Picos acentuados não universais (R >> h/2e2) em ambas as resistividades, local e não local, foram observados próximos ao CNP os quais diminuem rapidamente a medida que se afasta do CNP. Tal comportamento próximo ao CNP pode ser explicado usando o modelo de transporte de bordas (edge) e corpo (bulk), que inclui tanto os estados de borda como o corpo para a contribuição à corrente. O desvio dos valores da resistência de quarto terminais do valor quantizado (R >> h/2e2) em amostras macroscópicas com dimensões acima de algum mícron é um dos principais problemas no campo dos isolantes topológicos. Recentemente foi proposto um modelo por Vayrynen et al., onde tem sido considerado a influência de poças de carga, resultantes de distribuições de carga não homogêneas em isolantes topológicos 2d, na condutância de estados de borda helicoidal. Os estados de borda são acoplados por tunelamento a essas poças metálicas ou pontos quânticos. A permanência dos elétrons em pontos quânticos pode levar a um retroespalhamento inelástico significativo dentro da borda e modifica o transporte balístico. Portanto transporte balístico coerente é esperado somente na região entre poças, e o total de resistência de quatro terminais excede o valor quantizado. Introduzindo as interações elétron-elétron em sistemas de uma dimensão resulta em um liquido de Luttinger (LL). Os estados de borda helicoidais em isolantes topológicos 2d, podem ser tratados como um líquido de Luttinger ideal, uma vez que, naturalmente, aparecem em poços quânticos de HgTe. Entre as várias assinaturas específicas do comportamento do LL, como a dependência da temperatura, é importante se concentrar nas propriedades de não equilíbrio do LL. Em contraste com os líquidos de Fermi convencionais, nenhum estado excitado decairá ao estado de equilíbrio, caracterizado pela temperatura, na ausência de desordem. Medidas de elétron-aquecimento podem ser usadas para entender a física que governa os processos de relaxamento em LL. Nós temos realizado medidas de transporte não linear no CNP em isolantes topológicos 2d de HgTe. Este método, juntamente com a dependência da resistência com a temperatura, pode ser utilizado para determinar o mecanismo de relaxação da energia dos estados de borda helicoidais em QSHI. Nosso experimento falhou em confirmar as assinaturas especificas do comportamento do líquido de Luttinger. No entanto, o efeito de aquecimento de elétron pode ser descrito pelo mecanismo convencional de relaxamento de energia, esperado para espalhamento elétron-fônon. / This thesis present electronic transport properties of two-dimensional topological insulators (TI) based on HgTe/CdTe quantum wells. These heterostructures, in the band inverted regime, hosts a novel state known as the quantum spin Hall insulator. This state is identified as insulator in the bulk, but exhibits gapless conducting states at their edges which can be verified in transport experiments. Four-terminal resistance close to the quantized value has been observed in mesoscopic samples. However, for samples longer than 1 m, the resistance might be much higher than h/2e2 due to the presence of spin dephasing, inhomogeneity or disorder in the sample. This thesis address the problem of non-quantized resistance observed in macroscopic samples of dimensions longer than few microns. We report on the observation and a systematic investigation of local and nonlocal transport in HgTe quantum wells (8.0-8.3 nm) with inverted band structure corresponding to the quantum spin Hall insulating (QSHI) phase. The device MCT1 consists of three 4 m wide consecutive segments of different length (2 m, 8 m, 32 m), and seven voltage probes. The device MCT2 was fabricated with a lithographic length 6 m and width 5 m. Both devices are equipped with a top gate which allows tuning the carrier density of the device. Applying gate bias changes the carrier density transforming the quantum well conductivity from n-type to p-type via an intermediate phase, called the charge neutrality point (CNP). Non-universal (R >> h/2e2) peaks in both local and nonlocal resistivity were observed near the CNP which decreases rapidly going away from CNP. Such a behavior near CNP can be explained using the edge plus bulk transport model, which includes both the edge states and bulk contribution to the total current. Deviation of the four-terminal resistance from quantization (R >> h/2e2) in macroscopic samples, with dimensions above a few microns, is one of the major issue in the field of topological insulators. Recently a model was proposed by Vayrynen et al., where influence of charge puddles, resulting from inhomogeneous charge distribution in 2d topological insulators, on its helical edge conductance has been considered. The edge states are tunnel coupled to these metallic puddles or quantum dots. Electron´s dwelling in the quantum dot may lead to significant inelastic backscattering within the edge and modifies the ballistic transport. Therefore ballistic coherent transport is expected only in the region between the puddles, and the total four-terminal resistance exceeds the quantized value. Introducing electron-electron interactions in one-dimensional systems results in a Luttinger liquid (LL). The helical edge states in 2d topological insulator, can be treated as ideal Luttinger liquid, since it naturally appears in HgTe quantum wells. Among the various specific signatures of the LL behavior, such as temperature dependence, it is important to focus on non-equilibrium properties of LL. In contrast to conventional Fermi liquids, none of the excited state will decay to equilibrium state, characterized by temperature, in the absence of disorder. Electron-heating measurements can be used to understand the physics governing relaxation processes in LL. We have performed non-linear transport measurements at the CNP in HgTe based 2d topological insulators. This method together with temperature dependence of resistance can be used to determine the energy relaxation mechanism of the helical edge modes in QSHI. Our experiments fail to confirm the specific signatures of Luttinger liquid behavior. However, electron heating effect can be described by conventional energy relaxation mechanism, expected for electron-phonon interactions.

Quantum simulations, where controllable quantum systems are used to simulate other quantum systems, originally proposed by Richard Feynman, are one of the most remarkable applications of quantum information science. Compared to computation, quantum simulations require much less number of qubits for the m to be practical. In the work described in this thesis, we have performed a few quantum simulations of small quantum systems using Nuclear Magnetic Resonance(NMR) techniques. These simulations have been used to experimentally demonstrate the underlying interesting quantum protocols. All the experiments presented have been carried out using liquid-state or liquid crystal NMR. Numerical pulse optimization techniques have been utilized in some of the experiments, to achieve better control over the spin systems. The first chapter contains “Introduction” to quantum information processing, NMR, and numerical pulse optimization techniques. In chapter 2, we describe quantum simulation of a 3-spin Heisenberg-XY spin chain having only nearest neighbour interactions. Recently, spin chains having pre-engineered short-range interactions have been proposed to efficiently transfer quantum information between different parts of a quantum information processor. Other important proposals involving these spin chains include generating entangled states and universal quantum computation. However, such engineered interactions do not occur naturally in any system. In such a scenario, the experimental viability of these proposals can be tested by simulating the spin chains in other controllable quantum systems. In this work, we first theoretically study the time evolution of bipartite and tripartite entanglement measures for a 3-spin open ended XY spin chain. Then, by simulating the XY interactions in a 3-spin nuclear spin system, we experimentally generate, (i)a bipartite maximally(pseudo-)entangled state(Bell state) between end qubits, and(ii) multipartite(pseudo-)entangled states(Wand GHZ states),starting from separable pseudo-pure states. Bell state has been generated by using only the natural unitary evolution of the XY spin chain. W-state and GHZ-state have been generated by applying a single-qubit rotation to the second qubit, and a global rotation of all the three qubits respectively after the unitary evolution of the spin chain. In chapter 3, we simulate a 3-spin quantum transverse Ising spin system in a triangular configuration, and show that multipartite quantum correlations can be used to distinguish between the frustrated and non-frustrated regimes in the ground state of this spin system. The ground state of the spin system has been prepared by using adiabatic state preparation method. Gradient ascent pulse engineering technique has been utilized to efficiently realize the adiabatic evolution of the spin system. To analyse the experimental ground state of the system, we employ two different multipartite quantum correlation measures, generated from monogamy studies of bipartite quantum correlations. Chapter 4 contains a digital quantum simulation of the mirror inversion propagator corresponding to the time evolution of an XY spin chain. This simulation has been used to experimentally demonstrate the mirror inversion of quantum states, proposed by Albanese et al.[Phys.Rev.Lett.93,230502(2004)], by which entangled states can be transferred from one end of the chain to the other end. The experiments have been performed in a 5-qubit dipolar coupled nuclear spin system. For simulation, we make use of the recently proposed unitary operator decomposition algorithm along with the numerical pulse optimization techniques, which assisted in achieving high experimental fidelities. Chapter 5 contains a digital quantum simulation of the unitary propagator of a transverse Ising spin chain, which has been used to experimentally demonstrate the perfect state transfer protocol of Di Franco et al. [Phys.Rev.Lett.101,230502(2008)]. The importance of this protocol arises due to the fact that it achieves perfect state transfer from one end of the chain to the other end without the necessity of initializing the intermediate spins of the chain, whereas most of the previously proposed protocols require initialization. The experiments have been performed in a 3-spin nuclear spin system. The simulation has also been used to demonstrate the generation of a GHZ state.

Since the last few decades frustrated spin systems have attracted much interest. These studies are motivated by the rich variety of their unusual magnetic properties and potential applications. In this thesis, excitation spectra of the weakly coupled dimer system Ba3Cr2O8, the spin-1/2 chain material with distorted diamond structure Cu3(CO3)2(OH)2 (natural mineral azurite), and the quasi-twodimensional antiferromagnet with triangle spin structure Cs2CuBr4 have been studied by means of high-field electron spin resonance. Two pairs of gapped modes corresponding to transitions from a spin-singlet ground state to the first excited triplet state with zero-field energy gaps, of 19.1 and 27 K were observed in Ba3Cr2O8. The observation of ground-state excitations clearly indicates the presence of a non-secular term allowing these transitions. Our findings are of crucial importance for the interpretation of the field-induced transitions in this material (with critical fields Hc1 = 12.5 T and Hc2 = 23.6 T) in terms of the magnon Bose-Einstein condensation. The natural mineral azurite, Cu3(CO3)2(OH)2, has been studied in magnetic fields up to 50 T, revealing several modes not observed previously. Based on the obtained data, all three critical fields were identified. A substantial zero-field energy gap, Δ = 9.6 K, has been observed in Cs2CuBr4 above the ordering temperature. It is argued that contrary to the case for the isostructural Cs2CuCl4, the size of the gap can not be explained solely by the uniform Dzyaloshinskii-Moriya interaction, but it is rather the result of the geometrical frustration stabilizing the spin-disordered state in Cs2CuBr4 in the close vicinity of the quantum phase transition between a spiral magnetically ordered state and a 2D quantum spin liquid.

By numerical methods, two models of anisotropic mixed quantum spin chains, consisting of spins of two different sizes, Sa = 1/2 and Sb = 1 as well as Sb = 3/2, are studied with respect to their critical properties at quantum phase transitions in a selected region of parameter space. The quantum spin chains are made up of basecells of four spins, according to the structure Sa − Sa − Sb − Sb. They are described by the XXZ Hamiltonian, that extends the quantum Heisenberg model by a variable anisotropic exchange interaction. As additional control parameter, an alternating exchange constant between nearest-neighbour spins is introduced. Insight gained by complementary application of exact diagonalization and quantum Monte Carlo simulations, as well as appropriate methods of analysis, is embedded in the broad existing knowledge on homogeneous quantum spin chains. In anisotropic homogeneous quantum spin chains, there exist phase boundaries with continuously varying critical exponents, the Gaussian critical lines, along which, in addition to standard scaling relations, further extended scaling relations hold. Reweighting methods, also applied to improved quantum Monte Carlo estimators, and finite-size scaling analysis of simulation data deliver a wealth of numerical results confirming the existence of a Gaussian critical line also in the mixed spin models considered. Extrapolation of exact data offers, apart from confirmation of simulation data, furthermore, insight into the conformal operator content of the model with Sb = 1. / Mittels numerischer Methoden werden zwei Modelle anisotroper gemischter Quantenspinketten, bestehend aus Spins zweier unterschiedlicher Größen, Sa = 1/2 und Sb = 1 sowie Sb = 3/2, hinsichtlich ihrer kritischen Eigenschaften an Quanten-Phasenübergängen in einem ausgewählten Parameterbereich untersucht. Die Quantenspinketten sind aus Basiszellen zu vier Spins, gemäß der Struktur Sa − Sa − Sb − Sb, aufgebaut. Sie werden durch den XXZ Hamiltonoperator beschrieben, der das isotrope Quanten-Heisenberg Modell um eine variable anistrope Austauschwechselwirkung erweitert. Als zusätzlicher Kontrollparameter wird eine alterniernde Kopplungskonstante zwischen unmittelbar benachbarten Spins eingeführt. Die durch komplementäre Anwendung exakter Diagonalisierung und Quanten-Monte-Carlo Simulationen, sowie entsprechender Analyseverfahren, gewonnenen Erkenntnisse werden in das umfangreiche existierende Wissen über homogene Quantenspinketten eingebettet. Im Speziellen treten in anisotropen homogenen Quantenspinketten Phasengrenzen mit kontinuierlich variierenden kritischen Exponenten auf, die Gaußschen kritischen Linien, auf denen neben den herkömmlichen auch erweiterte Skalenrelationen Gültigkeit besitzen. Umgewichtungsmethoden, speziell auch angewandt auf verbesserte Quanten-Monte-Carlo Schätzer, und Endlichkeitsskalenanalyse von Simulationsdaten liefern eine Fülle von numerischen Ergebnissen, die das Auftreten der Gaußschen kritischen Linie auch in den untersuchten gemischten Quantenspinketten bestätigen. Die Extrapolation exakter Daten bietet, neben der Bestätigung der Simulationsdaten, darüber hinaus Einblick in einen Teil des konformen Operatorinhalts des Modells mit Sb = 1.

Dans la description à basse énergie de systèmes fortement corrélés, les champs de jauge peuvent émerger comme excitations collectives couplées à des quasiparticules fractionalisées. En particulier, certains aimants bidimensionnels dits frustrés sont décrits par un liquide de spin de Dirac comportant une symétrie de jauge U(1) compacte. La description infrarouge est donnée par une théorie conforme des champs, soit l'électrodynamique quantique en 2+1 dimensions avec 2N saveurs de fermions sans masse. Dans les aimants typiques, N=2 ou 4. L'aspect compact du champ de jauge implique également l'existence d'excitations topologiques, soit des instantons créés, dans ce contexte, par des opérateurs monopôles. Cette thèse porte sur les transitions de phase quantiques à partir d'un liquide de spin de Dirac et les propriétés des monopôles aux points critiques correspondants. Ces transitions sont induites en activant diverses interactions de type Gross-Neveu. Dans tous les cas à l'étude, la dimension d'échelle des monopôles est obtenue grâce à la correspondance état-opérateur et à un développement en 1/N. L'accent est d'abord mis sur une transition de confinement-déconfinement vers une phase antiferromagnétique décrite par la condensation d'un monopôle. Une levée de dégénérescence est observée au point critique alors que certaines dimensions d'échelle de monopôles sont réduites par rapport à leur valeur dans le liquide de spin de Dirac. Cette hiérarchie est caractérisée quantitativement en comparant les dimensions d'échelle dans des secteurs distincts du spin magnétique à l'ordre dominant en 1/N, puis qualitativement par une analyse en théorie des représentations. Des exposants critiques pour d'autres observables dans la théorie non compacte sont également obtenus. Enfin, deux transitions vers des liquides de spin topologiques, soit le liquide de spin chiral et le liquide de spin Z2, sont considérées. Les dimensions anormales des monopôles sont obtenues à l'ordre sous-dominant en 1/N. Ces résultats permettent de vérifier une dualité conjecturée avec un modèle bosonique et la valeur d'un coefficient universel pour les théories de jauge U(1) / In strongly correlated systems, gauge fields can emerge as collective excitations coupled to fractionalized quasiparticles. In particular, certain frustrated two-dimensional quantum magnets are described by a Dirac spin liquid which has a U(1) gauge symmetry. The infrared description is given by a conformal field theory, namely quantum electrodynamics in 2+1 dimensions with 2N flavours of massless fermions. In typical magnets, N=2 or 4. The compact aspect of the gauge field also implies the existence of topological excitations corresponding to instantons, which are created by monopole operators in this context. This thesis focuses on quantum phase transitions out of a Dirac spin liquid and the properties of monopoles at the corresponding critical points. These transitions are driven by activating various types of Gross-Neveu interactions. In all the cases studied, the scaling dimension of monopoles are obtained using the state-operator correspondence and a 1/N expansion. The confinement-deconfinement transition to an antiferromagnetic order produced by a monopole condensate is first studied. A degeneracy lifting is observed at the critical point, as certain monopoles have their scaling dimension reduced in comparison with the value in the Dirac spin liquid. This hierarchy is charactized quantitatively by comparing monopole scaling dimensions in distinct magnetic spin sector at leading-order in 1/N, and qualitatively by an analysis in representation theory. Critical exponents of various other operators are obtained in the non-compact model. Transitions to two topological spin liquids, namely a chiral spin liquid and a Z2 spin liquid, are also considered. Anomalous dimensions of monopoles are obtained at sub-leading order in 1/N. These results allow the verification of a conjectured duality with a bosonic model and the value of a universal coefficient in U(1) gauge theories.

Ce mémoire étudie le modèle quantique d’Ising-Kawasaki en une dimension. Cette chaîne quantique de spin-1/2 décrit la dynamique de Kawasaki hors d’équilibre d’une chaîne d’Ising classique couplée à un bain thermique. L’Hamiltonien est obtenu pour le cas général désor- donné avec des couplages d’Ising et champs magnétiques non-uniformes. Quand les champs magnétiques sont nuls, la chaîne de spin quantique est stochastique, et dépend des couplages d’Ising normalisés par la température du bain de chaleur. Dans le cas de couplages uniformes, nous donnons les états fondamentaux exacts de la chaîne de spin, ainsi que ses excitations à 1-magnon. Les solutions pour les spectres à deux magnons sont dérivées via une variante de l’Ansatz de Bethe. Dans le régime antiferromagnétique, les états de branche à deux magnons présentent un comportement complexe, notamment en ce qui concerne l’hybridation avec le continuum. L’analyse faite dans ce mémoire, combinée aux études précédentes, suggère que le système manifeste des dynamiques multiples à basse énergie, comme le montre la présence de plusieurs exposants critiques dynamiques. La distribution de l’espacement de l’ensemble des niveaux d’énergie est évaluée en fonction du couplage d’Ising. On conclut que le sys- tème est non-intégrable pour des paramètres génériques, ou de manière équivalente, que la dynamique classique hors équilibre correspondante est ergodique. / We study a quantum spin-1/2 chain that is dual to the non-equilibrium Kawasaki dynamics of a classical Ising chain coupled to a thermal bath. The Hamiltonian is obtained for the general disordered case with non-uniform Ising couplings. The quantum spin chain is stoquastic, and depends on the Ising couplings normalized by the bath’s temperature. Proceeding with uniform couplings, we give the exact groundstates of the gapless spin chain, as well as its single-magnon excitations. Solutions for the two-magnon spectra are derived via a Bethe Ansatz scheme. In the antiferromagnetic regime, the two-magnon branch states show intricate behavior, especially regarding hybridization with the continuum. Our analysis, when combined with previous studies, suggests that the system hosts multiple dynamics at low energy as seen via the presence of multiple dynamical critical exponents. Finally, we analyze the full energy level spacing distribution as a function of the Ising coupling. We conclude that the system is non-integrable for generic parameters, or equivalently, that the corresponding non-equilibrium classical dynamics are ergodic.

<p>In recent years, correlated magnetic phenomena have emerged as a unique resource for enabling alternative computing, memory, and sensing applications. This has led to the exploration of novel magnetic hybrid platforms with the promise of improved figures of merit over the state-of-the-art. In this dissertation, we delve into several example platforms where magnets interact with various other degrees of freedom, resulting in enhanced figures of merit and/or the emergence of novel functionalities.</p><p>First, we investigate the possibility of utilizing the collective resonant mode of nanomagnets to enhance the electric field sensitivity of quantum spin defects. While quantum systems have garnered significant attention in recent years for their extraordinary potential in information processing, their potential in the field of quantum sensing remains yet to be fully explored. Quantum systems, with their inherent fragility to external signals, can be harnessed as powerful tools to develop highly efficient sensors. In this dissertation, we explore the potential of a specific type of quantum sensor, namely the quantum spin defects as an electric field sensor, when integrated with a nanomagnet/piezoelectric composite multiferroic. This integration yields at least an order of magnitude enhancement in sensitivity, presenting a promising avenue for quantum sensing applications.</p><p>Next, we shift our focus towards harnessing magnetic correlation in the emerging class of atomically thin magnets known as van der Waals magnets. These magnets provide distinctive opportunities for controlling and exploiting magnetic correlations. Specifically, these platforms allow for tunable magnetic interactions by twisting two vertically adjacent layers of the magnet, features that are unique to van der Waals materials. By capitalizing on such twist degrees of freedom, we demonstrate the creation of twist-tunable nanoscale magnetic ground states. This capability opens up avenues for applications such as high-density memories and magnon crystals.</p><p>Interestingly, the same material platform also allows for exploiting magnetic correlation by controlling the local electrical environment. We uncover the symmetry-allowed spin-charge coupling mechanisms in the heterostructures of such magnets, a prediction that has received experimental support. Utilizing such an understanding, we propose a setup for the electrical generation of magnons. Magnons—the elementary excitation of spin waves—have garnered a lot of attention these days due to their potential to couple various diverse physical systems and in the field of low dissipation computing. Our findings offer a potential pathway towards the realization of magnon-based spintronic devices.</p>

Mon travail de thèse est consacré à l’étude de quelques aspects de la physique quantique des systèmes quantiques à N corps en interaction. Il est orienté vers l’étude des chaînes de spins quantiques. Je me suis intéressé à plusieurs questions relatives aux chaînes de spins quantiques, du point de vue numérique et analytique à la fois. J'aborde en particulier les questions relatives à la structure des fonctions d'onde, la forme de la densité d'états et les propriétés spectrales des Hamiltoniens de chaînes de spins. Dans un premier temps, je présenterais très rapidement les techniques numériques de base pour le calcul des vecteurs et valeurs propres des Hamiltonien de chaînes de spins. Les densités d’états des modèles quantiques constituent des quantités importantes et très simples qui permettent de caractériser les propriétés spectrales des systèmes avec un grand nombre de degrés de liberté. Alors que dans la limite thermodynamique, les densités d'états de la plupart des modèles intégrables sont bien décrites par une loi gaussienne, dans certaines limites de couplage de la chaîne de spins au champ magnétique et pour un nombre de spins N fini sur la chaîne, on observe l’apparition de pics dans la densité d’états. Je montrerais que la connaissance des deux premiers moments du Hamiltonien dans le sous-espace dégénéré associé à chaque pics donne une bonne approximation de la densité d’états. Dans un deuxième temps je m'intéresserais aux propriétés spectrales des Hamiltoniens de chaînes de spins quantiques. L’un des principal résultats sur la statistique spectrale des systèmes quantiques concerne le comportement universel des fluctuations des mesures telles que l’espacement entre valeurs propres consécutives. Ces fluctuations sont bien décrites par la théorie des matrices aléatoires mais la comparaison avec les prédictions de cette théorie nécessite généralement une opération sur le spectre du Hamiltonien appelée unfolding. Dans les problèmes quantiques de N corps, la taille de l’espace de Hilbert croît généralement exponentiellement avec le nombre de particules, entraînant un manque de données pour pouvoir faire une statistique. Ces limitations ont amené l’introduction d’une nouvelle mesure se passant de la procédure d’unfolding basée sur le rapport d’espacements successifs plutôt que les espacements. En suivant l’idée du “surmise” de Wigner pour le calcul de la distribution de l’espacement, je montre comment calculer une approximation de la distribution du rapport d’espacements dans les trois ensembles gaussiens invariants en faisant le calcul pour des matrices 3x3. Les résultats obtenus pour les différents ensembles de matrices aléatoires se sont révélés être en excellent accord avec les résultats numériques. Enfin je m’intéresserais à la structure des fonctions d’ondes fondamentales des modèles de chaînes de spins quantiques. Les fonctions d’onde constituent, avec le spectre en énergie, les objets fondamentaux des systèmes quantiques : leur structure est assez compliquée et n’est pas très bien comprise pour la plupart des systèmes à N corps. En raison de la croissance exponentielle de la taille de l’espace de Hilbert avec le nombre de particules, l’étude des vecteurs propres est une tâche très difficile, non seulement du point de vue analytique mais aussi du point de vue numérique. Je démontrerais en particulier que l’état fondamental de tous les modèles que nous avons étudiés est multifractal avec en général une dimension fractale non triviale. / My thesis is devoted to the study of some aspects of many body quantum interacting systems. In particular we focus on quantum spin chains. I have studied several aspects of quantum spin chains, from both numerical and analytical perspectives. I addressed especially questions related to the structure of eigenfunctions, the level densities and the spectral properties of spin chain Hamiltonians. In this thesis, I first present the basic numerical techniques used for the computation of eigenvalues and eigenvectors of spin chain Hamiltonians. Level densities of quantum models are important and simple quantities that allow to characterize spectral properties of systems with large number of degrees of freedom. It is well known that the level densities of most integrable models tend to the Gaussian in the thermodynamic limit. However, it appears that in certain limits of coupling of the spin chain to the magnetic field and for finite number of spins on the chain, one observes peaks in the level density. I will show that the knowledge of the first two moments of the Hamiltonian in the degenerate subspace associated with each peak give a good approximation to the level density. Next, I study the statistical properties of the eigenvalues of spin chain Hamiltonians. One of the main achievements in the study of the spectral statistics of quantum complex systems concerns the universal behaviour of the fluctuation of measure such as the distribution of spacing between two consecutive eigenvalues. These fluctuations are very well described by the theory of random matrices but the comparison with the theoretical prediction generally requires a transformation of the spectrum of the Hamiltonian called the unfolding procedure. For many-body quantum systems, the size of the Hilbert space generally grows exponentially with the number of particles leading to a lack of data to make a proper statistical study. These constraints have led to the introduction of a new measure free of the unfolding procedure and based on the ratio of consecutive level spacings rather than the spacings themselves. This measure is independant of the local level density. By following the Wigner surmise for the computation of the level spacing distribution, I obtained approximation for the distribution of the ratio of consecutive level spacings by analyzing random 3x3 matrices for the three canonical ensembles. The prediction are compared with numerical results showing excellent agreement. Finally, I investigate eigenfunction statistics of some canonical spin-chain Hamiltonians. Eigenfunctions together with the energy spectrum are the fundamental objects of quantum systems: their structure is quite complicated and not well understood. Due to the exponential growth of the size of the Hilbert space, the study of eigenfunctions is a very difficult task from both analytical and numerical points of view. I demonstrate that the groundstate eigenfunctions of all canonical models of spin chain are multifractal, by computing numerically the Rényi entropy and extrapolating it to obtain the multifractal dimensions.