Due to recent progress in theory and the growing number of physical realizations, low-dimensional quantum magnets continue to receive a considerable amount of attention. They serve as model systems for investigating numerous physical phenomena in spin systems with cooperative ground states, including the field-induced evolution of the ground-state properties and the corresponding rearrangement of their low-energy excitation spectra. This work is devoted to systematic studies of recently synthesized low-dimensional quantum spin systems by means of multi-frequency high-field electron spin resonance (ESR) investigations. In the spin- 1/2 chain compound (C6H9N2)CuCl3 [known as (6MAP)CuCl3] the striking incompatibility with a simple uniform S = 1/2 Heisenberg chain model employed previously is revealed. The observed ESR mode is explained in terms of a recently developed theory, revealing the important role of the alternation and next-nearest-neighbor interactions in this compound. The excitations spectrum in copper pyrimidine dinitrate [PM·Cu(NO3)2(H2O)2]n, an S = 1/2 antiferromagnetic chain material with alternating g-tensor and Dzyaloshinskii-Moriya interaction, is probed in magnetic fields up to 63 T. To study the high field behavior of the field-induced energy gap in this material, a multi-frequency pulsed-field ESR spectrometer is built. Pronounced changes in the frequency-field dependence of the magnetic excitations are observed in the vicinity of the saturation field, B ∼ Bs = 48.5 T. ESR results clearly indicate a transition from the soliton-breather to a spin-polarized state with magnons as elementary excitations. Experimental data are compared with results of density matrix renormalization group calculations; excellent agreement is found. ESR studies of the spin-ladder material (C5H12N)2CuBr4 (known as BPCB) completes the determination of the full spin Hamiltonian of this compound. ESR results provide a direct evidence for a pronounced anisotropy in this compound, that is in contrast to fully isotropic spin-ladder model employed previously for BPCB. Our observations can be of particular importance for describing the rich temperature-field phase diagram of this material. The frequency-field diagram of magnetic excitations in the quasi-two dimensional S = 1/2 compound [Cu(C4H4N2)2(HF2)]PF6 in the AFM-ordered state is studied. The AFM gap is observed directly. Using high-field magnetization and ESR results, parameters of the effective spin-Hamiltonian (exchange interaction, anisotropy and g-factor) are obtained and compared with those estimated from thermodynamic properties of this compound.

Spatial confinement of electrons and their interactions as well as confinement of the spin dimensionality often yield drastic changes of the electronic and magnetic properties of solids. Novel quantum transport and optical phenomena, involving electronic spin degrees of freedom in semiconductor heterostructures, as well as a rich variety of exotic quantum ground states and magnetic excitations in complex transition metal oxides that arise upon such confinements, belong therefore to topical problems of contemporary condensed matter physics. In this work electron spin systems in reduced dimensions are studied with Electron Spin Resonance (ESR) spectroscopy, a method which can provide important information on the energy spectrum of the spin states, spin dynamics, and magnetic correlations. The studied systems include quasi onedimensional spin chain materials based on transition metals Cu and Ni. Another class of materials are semiconductor heterostructures made of Si and Ge. Part I deals with the theoretical background of ESR and the description of the experimental ESR setups used which have been optimized for the purposes of the present work. In particular, the development and implementation of axial and transverse cylindrical resonant cavities for high-field highfrequency ESR experiments is discussed. The high quality factors of these cavities allow for sensitive measurements on μm-sized samples. They are used for the investigations on the spin-chain materials. The implementation and characterization of a setup for electrical detected magnetic resonance is presented. In Part II ESR studies and complementary results of other experimental techniques on two spin chain materials are presented. The Cu-based material Linarite is investigated in the paramagnetic regime above T > 2.8 K. This natural crystal constitutes a highly frustrated spin 1/2 Heisenberg chain with ferromagnetic nearest-neighbor and antiferromagnetic next-nearestneighbor interactions. The ESR data reveals that the significant magnetic anisotropy is due to anisotropy of the g-factor. Quantitative analysis of the critical broadening of the linewidth suggest appreciable interchain and interlayer spin correlations well above the ordering temperature. The Ni-based system is an organic-anorganic hybrid material where the Ni2+ ions possessing the integer spin S = 1 are magnetically coupled along one spatial direction. Indeed, the ESR study reveals an isotropic spin-1 Heisenberg chain in this system which unlike the Cu half integer spin-1/2 chain is expected to possess a qualitatively different non-magnetic singlet ground state separated from an excited magnetic state by a so-called Haldane gap. Surprisingly, in contrast to the expected Haldane behavior a competition between a magnetically ordered ground state and a potentially gapped state is revealed. In Part III investigations on SiGe/Si quantum dot structures are presented. The ESR investigations reveal narrowlines close to the free electron g-factor associated with electrons on the quantum dots. Their dephasing and relaxation times are determined. Manipulations with sub-bandgap light allow to change the relative population between the observed states. On the basis of extensive characterizations, strain, electronic structure and confined states on the Si-based structures are modeled with the program nextnano3. A qualitative model, explaining the energy spectrum of the spin states is proposed.

Dans cette thèse, la connexion entre certaines structures algébriques récentes (algèbres tridiagonales, algèbre q-Onsager, algèbres q-Onsager généralisées), la théorie des représentations (paire tridiagonale, paire de Leonard, polynômes orthogonaux), certaines des propriétés de ces algèbres et l’analyse de modèles intégrables quantiques sur le réseau (la chaîne de spin XXZ ouverte aux racines de l’unité) est considérée. / In this thesis, the connection between recently introduced algebraic structures (tridiagonal algebra, q-Onsager algebra, generalized q-Onsager algebras), related representation theory (tridiagonal pair, Leonard pair, orthogonal polynomials), some properties of these algebras and the analysis of related quantum integrable models on the lattice (the XXZ open spin chain at roots of unity) is considered.

<div><div>In this work we use and develop Riemann-Hilbert techniques to study the asymptotic behavior of structured determinants. In chapter one we will review the main underlying</div><div>definitions and ideas which will be extensively used throughout the thesis. Chapter two is devoted to the asymptotic analysis of Hankel determinants with Laguerre-type and Jacobi-type potentials with Fisher-Hartwig singularities. In chapter three we will propose a Riemann-Hilbert problem for Toeplitz+Hankel determinants. We will then analyze this Riemann-Hilbert problem for a certain family of Toeplitz and Hankel symbols. In Chapter four we will study the asymptotics of a certain bordered-Toeplitz determinant which is related to the next-to-diagonal correlations of the anisotropic Ising model. The analysis is based upon relating the bordered-Toeplitz determinant to the solution of the Riemann-Hilbert problem associated to pure Toeplitz determinants. Finally in chapter ve we will study the emptiness formation probability in the XXZ-spin 1/2 Heisenberg chain, or equivalently, the asymptotic analysis of the associated Fredholm determinant.</div></div>

Due to recent progress in theory and the growing number of physical realizations, low-dimensional quantum magnets continue to receive a considerable amount of attention. They serve as model systems for investigating numerous physical phenomena in spin systems with cooperative ground states, including the field-induced evolution of the ground-state properties and the corresponding rearrangement of their low-energy excitation spectra. This work is devoted to systematic studies of recently synthesized low-dimensional quantum spin systems by means of multi-frequency high-field electron spin resonance (ESR) investigations. In the spin- 1/2 chain compound (C6H9N2)CuCl3 [known as (6MAP)CuCl3] the striking incompatibility with a simple uniform S = 1/2 Heisenberg chain model employed previously is revealed. The observed ESR mode is explained in terms of a recently developed theory, revealing the important role of the alternation and next-nearest-neighbor interactions in this compound. The excitations spectrum in copper pyrimidine dinitrate [PM·Cu(NO3)2(H2O)2]n, an S = 1/2 antiferromagnetic chain material with alternating g-tensor and Dzyaloshinskii-Moriya interaction, is probed in magnetic fields up to 63 T. To study the high field behavior of the field-induced energy gap in this material, a multi-frequency pulsed-field ESR spectrometer is built. Pronounced changes in the frequency-field dependence of the magnetic excitations are observed in the vicinity of the saturation field, B ∼ Bs = 48.5 T. ESR results clearly indicate a transition from the soliton-breather to a spin-polarized state with magnons as elementary excitations. Experimental data are compared with results of density matrix renormalization group calculations; excellent agreement is found. ESR studies of the spin-ladder material (C5H12N)2CuBr4 (known as BPCB) completes the determination of the full spin Hamiltonian of this compound. ESR results provide a direct evidence for a pronounced anisotropy in this compound, that is in contrast to fully isotropic spin-ladder model employed previously for BPCB. Our observations can be of particular importance for describing the rich temperature-field phase diagram of this material. The frequency-field diagram of magnetic excitations in the quasi-two dimensional S = 1/2 compound [Cu(C4H4N2)2(HF2)]PF6 in the AFM-ordered state is studied. The AFM gap is observed directly. Using high-field magnetization and ESR results, parameters of the effective spin-Hamiltonian (exchange interaction, anisotropy and g-factor) are obtained and compared with those estimated from thermodynamic properties of this compound.

Spatial confinement of electrons and their interactions as well as confinement of the spin dimensionality often yield drastic changes of the electronic and magnetic properties of solids. Novel quantum transport and optical phenomena, involving electronic spin degrees of freedom in semiconductor heterostructures, as well as a rich variety of exotic quantum ground states and magnetic excitations in complex transition metal oxides that arise upon such confinements, belong therefore to topical problems of contemporary condensed matter physics. In this work electron spin systems in reduced dimensions are studied with Electron Spin Resonance (ESR) spectroscopy, a method which can provide important information on the energy spectrum of the spin states, spin dynamics, and magnetic correlations. The studied systems include quasi onedimensional spin chain materials based on transition metals Cu and Ni. Another class of materials are semiconductor heterostructures made of Si and Ge. Part I deals with the theoretical background of ESR and the description of the experimental ESR setups used which have been optimized for the purposes of the present work. In particular, the development and implementation of axial and transverse cylindrical resonant cavities for high-field highfrequency ESR experiments is discussed. The high quality factors of these cavities allow for sensitive measurements on μm-sized samples. They are used for the investigations on the spin-chain materials. The implementation and characterization of a setup for electrical detected magnetic resonance is presented. In Part II ESR studies and complementary results of other experimental techniques on two spin chain materials are presented. The Cu-based material Linarite is investigated in the paramagnetic regime above T > 2.8 K. This natural crystal constitutes a highly frustrated spin 1/2 Heisenberg chain with ferromagnetic nearest-neighbor and antiferromagnetic next-nearestneighbor interactions. The ESR data reveals that the significant magnetic anisotropy is due to anisotropy of the g-factor. Quantitative analysis of the critical broadening of the linewidth suggest appreciable interchain and interlayer spin correlations well above the ordering temperature. The Ni-based system is an organic-anorganic hybrid material where the Ni2+ ions possessing the integer spin S = 1 are magnetically coupled along one spatial direction. Indeed, the ESR study reveals an isotropic spin-1 Heisenberg chain in this system which unlike the Cu half integer spin-1/2 chain is expected to possess a qualitatively different non-magnetic singlet ground state separated from an excited magnetic state by a so-called Haldane gap. Surprisingly, in contrast to the expected Haldane behavior a competition between a magnetically ordered ground state and a potentially gapped state is revealed. In Part III investigations on SiGe/Si quantum dot structures are presented. The ESR investigations reveal narrowlines close to the free electron g-factor associated with electrons on the quantum dots. Their dephasing and relaxation times are determined. Manipulations with sub-bandgap light allow to change the relative population between the observed states. On the basis of extensive characterizations, strain, electronic structure and confined states on the Si-based structures are modeled with the program nextnano3. A qualitative model, explaining the energy spectrum of the spin states is proposed.:Abstract i Contents iii List of Figures vi List of Tables viii 1 Preface 1 I Background and Experimental 5 2 Principles of ESR 7 2.1 The Resonance Phenomenon . . . . . . . . . . . . . . . . . . . 7 2.2 ESR Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.2.1 The g -factor . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.2.2 Relaxation Times . . . . . . . . . . . . . . . . . . . . . . 12 2.2.3 Lineshape Properties . . . . . . . . . . . . . . . . . . . . 13 2.3 Effective Spin Hamiltonian . . . . . . . . . . . . . . . . . . . . . 15 2.4 Spin-Orbit Coupling . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.5 d-electrons in a Crystal Field . . . . . . . . . . . . . . . . . . . . 17 2.6 Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.6.1 Dipolar Coupling . . . . . . . . . . . . . . . . . . . . . . 23 2.6.2 Exchange Interaction . . . . . . . . . . . . . . . . . . . . 23 2.6.3 Superexchange . . . . . . . . . . . . . . . . . . . . . . . 24 2.6.4 Symmetric Anisotropic Exchange . . . . . . . . . . . . 25 2.6.5 Antisymmetric Anisotropic Exchange . . . . . . . . . . 25 2.6.6 Hyperfine Interaction . . . . . . . . . . . . . . . . . . . 26 3 Experimental 27 3.1 Setup for Experiments at 10GHz . . . . . . . . . . . . . . . . . 27 3.2 Implementation of an EDMR Setup . . . . . . . . . . . . . . . . 29 3.2.1 Basic Characterization . . . . . . . . . . . . . . . . . . . 31 3.3 High Frequency Setup . . . . . . . . . . . . . . . . . . . . . . . . 31 3.3.1 MillimeterWave Vector Network Analyzer . . . . . . . 33 3.3.2 Waveguides and Cryostats . . . . . . . . . . . . . . . . . 34 3.4 Development of the Resonant Cavity Setup . . . . . . . . . . . 35 3.4.1 Mode Propagation . . . . . . . . . . . . . . . . . . . . . 38 3.4.2 Resonant CavityModes . . . . . . . . . . . . . . . . . . 40 3.4.3 Resonant Cavity Design . . . . . . . . . . . . . . . . . . 41 3.4.4 Resonant Cavity Sample Stick . . . . . . . . . . . . . . . 45 3.4.5 Experimental Characterization . . . . . . . . . . . . . . 47 3.4.6 Performing an ESR Experiment . . . . . . . . . . . . . . 53 II Quasi One-Dimensional Spin-Chains 57 4 Motivation 59 5 Quasi One-Dimensional Systems 61 5.1 Magnetic Order and Excitations . . . . . . . . . . . . . . . . . . 63 5.2 Competing Interactions . . . . . . . . . . . . . . . . . . . . . . . 64 5.3 Haldane Spin Chain . . . . . . . . . . . . . . . . . . . . . . . . . 66 6 Linarite 69 6.1 Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 6.2 Magnetization and ESR . . . . . . . . . . . . . . . . . . . . . . . 71 6.3 NMR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 6.4 Summary and Conclusion . . . . . . . . . . . . . . . . . . . . . 81 6.5 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 7 The Ni-hybrid NiCl3C6H5CH2CH2NH3 83 7.1 Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 7.2 Susceptibility andMagnetization . . . . . . . . . . . . . . . . . 85 7.3 ESR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 7.4 Further Investigations . . . . . . . . . . . . . . . . . . . . . . . . 95 7.5 Summary and Conclusion . . . . . . . . . . . . . . . . . . . . . 96 8 Summary 99 III SiGe Nanostructures 101 9 Motivation 103 10 SiGe Semiconductor Nanostructures 107 10.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 10.1.1 Silicon and Germanium . . . . . . . . . . . . . . . . . . 107 10.1.2 Epitaxial Growth of SiGe Heterostructures . . . . . . . 109 10.1.3 Strain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 10.1.4 Band Deformation . . . . . . . . . . . . . . . . . . . . . 112 10.2 Sample Structure and Characterization . . . . . . . . . . . . . 114 11 Modelling of SiGe/Si Heterostructures 119 11.1 Program Structure . . . . . . . . . . . . . . . . . . . . . . . . . . 120 11.2 Implementation of Si/Ge System . . . . . . . . . . . . . . . . . 121 11.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 11.3.1 Single Quantum Dot . . . . . . . . . . . . . . . . . . . . 123 11.3.2 Multiple Quantum Dots . . . . . . . . . . . . . . . . . . 127 11.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 11.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 12 ESR Experiments on Si/SiGe Quantum Dots 135 12.1 ESR on Si Structures . . . . . . . . . . . . . . . . . . . . . . . . . 135 12.2 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . 137 12.2.1 Samples grown at 600◦C . . . . . . . . . . . . . . . . . . 138 12.2.2 Samples grown at 700◦C . . . . . . . . . . . . . . . . . . 139 12.2.3 T1-Relaxation Time . . . . . . . . . . . . . . . . . . . . . 143 12.2.4 Effect of Illumination . . . . . . . . . . . . . . . . . . . . 145 12.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 12.3.1 Quantum Dots . . . . . . . . . . . . . . . . . . . . . . . 149 12.3.2 Assignment of ESR Lines . . . . . . . . . . . . . . . . . . 150 12.3.3 Relaxation Times . . . . . . . . . . . . . . . . . . . . . . 153 12.3.4 Donors in Heterostructures . . . . . . . . . . . . . . . . 153 12.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156 13 Summary and Outlook 157 Bibliography 163 Acknowledgements 176

In der eingereichten Dissertation wird eine Reihe von niederdimensionalen und frustrierten magnetischen Systemen mit Hilfe der Elektronenspinresonanz (ESR) untersucht, um deren magnetische Eigenschaften und Wechselwirkungen zu charakterisieren. Sowohl niederdimensionale als auch frustrierte Systeme können exotische magnetische Phänomene zeigen, da es in beiden Fällen trotz starker magnetischer Korrelationen zu einer Unterdrückung von konventioneller langreichweitiger magnetischer Ordnung kommen kann. Auf der anderen Seite sind zweidimensionale Systeme wie Graphen und die damit verwandten topologischen Isolatoren interessant für Anwendungen in der Spintronik oder in Quantencomputern. Über das Einbringen von magnetischer Ordnung soll dabei die Kontrolle über den Spin von Elektronen erlangt werden. Es werden quasieindimensionale Spinketten in Cu(py)2Br2 untersucht, die ein gutes Modellsysteme für den Vergleich mit exakten theoretischen Berechnungen darstellen. Durch eingehende ESR-Messungen ist es gelungen, ein Modell für die Ausrichtung der Anisotropieachse zu entwickeln, die senkrecht zur Kettenachse steht. Zusätzlich zum g-Tensor konnten durch Magnetisierungsmessungen das Austauschintegral und dessen Anisotropie bestimmt werden. Die Austauschwechselwirkung kann über die Substitution von Br- mit Cl-Ionen in Cu(py)2(Cl1-xBrx)2 gezielt variiert werden. Des Weiteren wird eine kombinierte Studie aus STM- und ESR-Untersuchungen an monolagigem Graphen mit induzierten Fehlstellen vorgestellt. Es wurden Defekte durch den Beschuss mit Ar-Ionen in Graphen kontrolliert hergestellt, deren lokale elektronische Eigenschaften sich mit STM- und STS-Messungen charakte-risieren lassen. Mit ESR-Messungen konnte gezeigt werden, dass die an den einzelnen Fehlstellen lokalisierten magnetischen Momente eine dominant antiferromagnetische Austauschwechselwirkung besitzen. Die Charakterisierung der magnetischen Wechselwirkungen zwischen lokalisierten Momenten stand auch für den mit Mn dotierten topologischen Isolator Bi2Te3 im Vordergrund, welcher einen ferromagnetischen Phasenübergang bei tiefen Temperaturen zeigt. Anhand des mit ESR beobachteten Korringa-Verhaltens wurde bewiesen, dass die lokalisierten Mn-Spins an leitende Bänder gekoppelt sind und die ferromagnetische Ordnung folglich per RKKY-Wechselwirkung vermittelt wird. Es wurden kurzreichweitige magnetische Korrelationen in einem ausgedehnten Temperaturbereich oberhalb der Ordnungstemperatur beobachtet, die Hinweise auf einen zweidimensionalen Charakter zeigen. Ausgedehnte Temperaturbereiche mit kurzreichweitigen Korrelationen werden ebenfalls in den untersuchten magnetisch frustrierten Materialien beobachtet. In einer kombinierten Studie aus HF-ESR, NMR und µSR wird die Spindynamik in CoAl2O4 charakterisiert, in dem moderate Unordnung zu einem Verschwimmen der Phasengrenze zwischen Neél-Ordnung und einer Spinflüssigkeit mit spiralförmigen Korrelationen führt. Außerdem werden zwei Vertreter aus der Klasse der Swedenborgite behandelt, in denen die Spinstruktur in YBaCo4O7 durch Substitution modifiziert wird. Ziel ist die Entkopplung der enthaltenen Kagome-Schichten, welche ein zweidimensionales frustriertes System darstellen. In den vorgestellten HF-ESR- und NMR-Messungen beobachtet man ein Spinglasverhalten für YBaCo3AlO7, das aus der Unordnung bei der Besetzung der Gitterplätze resultiert. In YBaCo3FeO7 ist die Unordnung geringer und mit ESR-Untersuchungen konnte gezeigt werden, dass es zu einer effektiven Entkopplung der Fe-Spins zwischen den Kagome-Schichten kommt.

Cette thèse porte sur les phénomènes critiques survenant dans les modèles bidimensionnels sur réseau. Les résultats sont l'objet de deux articles : le premier porte sur la mesure d'exposants critiques décrivant des objets géométriques du réseau et, le second, sur la construction d'idempotents projetant sur des modules indécomposables de l'algèbre de Temperley-Lieb pour la chaîne de spins XXZ. Le premier article présente des expériences numériques Monte Carlo effectuées pour une famille de modèles de boucles en phase diluée. Baptisés "dilute loop models (DLM)", ceux-ci sont inspirés du modèle O(n) introduit par Nienhuis (1990). La famille est étiquetée par les entiers relativement premiers p et p' ainsi que par un paramètre d'anisotropie. Dans la limite thermodynamique, il est pressenti que le modèle DLM(p,p') soit décrit par une théorie logarithmique des champs conformes de charge centrale c(\kappa)=13-6(\kappa+1/\kappa), où \kappa=p/p' est lié à la fugacité du gaz de boucles \beta=-2\cos\pi/\kappa, pour toute valeur du paramètre d'anisotropie. Les mesures portent sur les exposants critiques représentant la loi d'échelle des objets géométriques suivants : l'interface, le périmètre externe et les liens rouges. L'algorithme Metropolis-Hastings employé, pour lequel nous avons introduit de nombreuses améliorations spécifiques aux modèles dilués, est détaillé. Un traitement statistique rigoureux des données permet des extrapolations coïncidant avec les prédictions théoriques à trois ou quatre chiffres significatifs, malgré des courbes d'extrapolation aux pentes abruptes. Le deuxième article porte sur la décomposition de l'espace de Hilbert \otimes^nC^2 sur lequel la chaîne XXZ de n spins 1/2 agit. La version étudiée ici (Pasquier et Saleur (1990)) est décrite par un hamiltonien H_{XXZ}(q) dépendant d'un paramètre q\in C^\times et s'exprimant comme une somme d'éléments de l'algèbre de Temperley-Lieb TL_n(q). Comme pour les modèles dilués, le spectre de la limite continue de H_{XXZ}(q) semble relié aux théories des champs conformes, le paramètre q déterminant la charge centrale. Les idempotents primitifs de End_{TL_n}\otimes^nC^2 sont obtenus, pour tout q, en termes d'éléments de l'algèbre quantique U_qsl_2 (ou d'une extension) par la dualité de Schur-Weyl quantique. Ces idempotents permettent de construire explicitement les TL_n-modules indécomposables de \otimes^nC^2. Ceux-ci sont tous irréductibles, sauf si q est une racine de l'unité. Cette exception est traitée séparément du cas où q est générique. Les problèmes résolus par ces articles nécessitent une grande variété de résultats et d'outils. Pour cette raison, la thèse comporte plusieurs chapitres préparatoires. Sa structure est la suivante. Le premier chapitre introduit certains concepts communs aux deux articles, notamment une description des phénomènes critiques et de la théorie des champs conformes. Le deuxième chapitre aborde brièvement la question des champs logarithmiques, l'évolution de Schramm-Loewner ainsi que l'algorithme de Metropolis-Hastings. Ces sujets sont nécessaires à la lecture de l'article "Geometric Exponents of Dilute Loop Models" au chapitre 3. Le quatrième chapitre présente les outils algébriques utilisés dans le deuxième article, "The idempotents of the TL_n-module \otimes^nC^2 in terms of elements of U_qsl_2", constituant le chapitre 5. La thèse conclut par un résumé des résultats importants et la proposition d'avenues de recherche qui en découlent. / This thesis is concerned with the study of critical phenomena for two-dimensional models on the lattice. Its results are contained in two articles: A first one, devoted to measuring geometric exponents, and a second one to the construction of idempotents for the XXZ spin chain projecting on indecomposable modules of the Temperley-Lieb algebra. Monte Carlo experiments, for a family of loop models in their dilute phase, are presented in the first article. Coined "dilute loop models (DLM)", this family is based upon an O(n) model introduced by Nienhuis (1990). It is defined by two coprime integers p,p' and an anisotropy parameter. In the continuum limit, DLM(p,p') is expected to yield a logarithmic conformal field theory of central charge c(\kappa)=13-6(\kappa+1/\kappa), where the ratio \kappa=p/p' is related to the loop gas fugacity \beta=-2\cos\pi/\kappa. Critical exponents pertaining to valuable geometrical objects, namely the hull, external perimeter and red bonds, were measured. The Metropolis-Hastings algorithm, as well as several methods improving its efficiency, are presented. Despite the extrapolation of curves presenting large slopes, values as close as three to four digits from the theoretical predictions were attained through rigorous statistical analysis. The second article describes the decomposition of the XXZ spin chain Hilbert space \otimes^nC^2 using idempotents. The model of interest (Pasquier & Saleur (1990)) is described by a parameter-dependent Hamiltonian H_{XXZ}(q), q\in C^\times, expressible as a sum of elements of the Temperley-Lieb algebra TL_n(q). The spectrum of H_{XXZ}(q) in the continuum limit is also believed to be related to conformal field theories whose central charge is set by q. Using the quantum Schur-Weyl duality, an expression for the primitive idempotents of End_{TL_n}\otimes^nC^2, involving U_qsl_2 elements, is obtained. These idempotents allow for the explicit construction of the indecomposable TL_n-modules of \otimes^nC^2, all of which are irreducible except when q is a root of unity. This case, and the case where q is generic, are treated separately. Since a wide variety of results and tools are required to tackle the problems stated above, this thesis contains many introductory chapters. Its layout is as follows. The first chapter introduces theoretical concepts common to both articles, in particular an overview of critical phenomena and conformal field theory. Before proceeding to the article entitled \emph{Geometric Exponents of Dilute Loop Models} constituting Chapter 3, the second chapter deals briefly with logarithmic conformal fields, Schramm-Loewner evolution and the Metropolis-Hastings algorithm. The fourth chapter defines some algebraic concepts used in the second article, "The idempotents of the TL_n-module \otimes^nC^2 in terms of elements of U_qsl_2" of Chapter 5. A summary of the main results, as well as paths to unexplored questions, are suggested in a final chapter.

Within the last decades low-dimensional frustrated quantum spin systems have attracted great interest in the field of modern research. In these systems a competition of various magnetic interactions takes place, leading to an energetically degenerated magnetic ground state, and thus to the occurrence of exotic, unconventional physical properties at low temperatures. This thesis focuses on the quasi one-dimensional frustrated spin chain system linarite, PbCuSO4(OH)2. In this compound the basic building blocks are CuO4 plaquettes which are connected to each other along one crystallographic direction, analogue to a chain. The frustration in linarite is established due to the competition between the magnetic interactions. The nearest-neighbor magnetic spins are coupled ferromagnetically along the chain via a coupling constant J1, while the next-nearest neighbors are coupled antiferromagnetically via a coupling constant J2. For this configuration it is not possible to satisfy all magnetic couplings simultaneously, hence the system is magnetically frustrated. In this work, comprehensive thermodynamic and nuclear magnetic resonance (NMR) studies demonstrate that linarite is one of the richest and most fascinating compounds in the class of low-dimensional frustrated magnets. By means of susceptibility, magnetization, specific heat, magnetocaloric effect, magnetostriction, and thermal-expansion measurements a rich magnetic phase diagram could be mapped out below a temperature of 2.8 K. The phase diagram contains five different magnetic regions/phases for an external magnetic field pointing along the chain direction. Based on the thermodynamic studies it was possible to calculate the exchange integrals within the frustrated J1-J2 model and extensions of it by using various theoretical approaches. The magnetic microscopic nature of the different long-range magnetic phases present in linarite were investigated by NMR measurements and by collaborative neutron scattering experiments. The ground state (phase I) is identified as an incommensurate elliptical helical structure. Via a theoretical modelling the 1H-NMR spectrum of the ground state could be explained, revealing a rearrangement of the zero-field structure in an external magnetic field of 2.0 T used for the NMR studies. By further increasing the external field the system undergoes a complex spin flop transition in two steps (phase I - phase III - phase IV). In phase III a phase separation takes place where one part of the spins form a circular spiral structure while the remaining fraction form a simple antiferromagnetic structure. In phase IV the remaining circular spiral structure vanishes, so that all spins collectively form the antiferromagnetic collinear phase. The most peculiar physical properties studied in this thesis take place in region V at high fields, showing only tiny features in the thermodynamic properties. The magnetic spins in region V form a sine-wave modulated spin-density structure as identified via NMR and neutron investigations. It is discussed whether region V is related to a multipolar phase or if the spin-density wave structure could possibly coexist with such a phase.

Cette thèse porte sur les phénomènes critiques survenant dans les modèles bidimensionnels sur réseau. Les résultats sont l'objet de deux articles : le premier porte sur la mesure d'exposants critiques décrivant des objets géométriques du réseau et, le second, sur la construction d'idempotents projetant sur des modules indécomposables de l'algèbre de Temperley-Lieb pour la chaîne de spins XXZ. Le premier article présente des expériences numériques Monte Carlo effectuées pour une famille de modèles de boucles en phase diluée. Baptisés "dilute loop models (DLM)", ceux-ci sont inspirés du modèle O(n) introduit par Nienhuis (1990). La famille est étiquetée par les entiers relativement premiers p et p' ainsi que par un paramètre d'anisotropie. Dans la limite thermodynamique, il est pressenti que le modèle DLM(p,p') soit décrit par une théorie logarithmique des champs conformes de charge centrale c(\kappa)=13-6(\kappa+1/\kappa), où \kappa=p/p' est lié à la fugacité du gaz de boucles \beta=-2\cos\pi/\kappa, pour toute valeur du paramètre d'anisotropie. Les mesures portent sur les exposants critiques représentant la loi d'échelle des objets géométriques suivants : l'interface, le périmètre externe et les liens rouges. L'algorithme Metropolis-Hastings employé, pour lequel nous avons introduit de nombreuses améliorations spécifiques aux modèles dilués, est détaillé. Un traitement statistique rigoureux des données permet des extrapolations coïncidant avec les prédictions théoriques à trois ou quatre chiffres significatifs, malgré des courbes d'extrapolation aux pentes abruptes. Le deuxième article porte sur la décomposition de l'espace de Hilbert \otimes^nC^2 sur lequel la chaîne XXZ de n spins 1/2 agit. La version étudiée ici (Pasquier et Saleur (1990)) est décrite par un hamiltonien H_{XXZ}(q) dépendant d'un paramètre q\in C^\times et s'exprimant comme une somme d'éléments de l'algèbre de Temperley-Lieb TL_n(q). Comme pour les modèles dilués, le spectre de la limite continue de H_{XXZ}(q) semble relié aux théories des champs conformes, le paramètre q déterminant la charge centrale. Les idempotents primitifs de End_{TL_n}\otimes^nC^2 sont obtenus, pour tout q, en termes d'éléments de l'algèbre quantique U_qsl_2 (ou d'une extension) par la dualité de Schur-Weyl quantique. Ces idempotents permettent de construire explicitement les TL_n-modules indécomposables de \otimes^nC^2. Ceux-ci sont tous irréductibles, sauf si q est une racine de l'unité. Cette exception est traitée séparément du cas où q est générique. Les problèmes résolus par ces articles nécessitent une grande variété de résultats et d'outils. Pour cette raison, la thèse comporte plusieurs chapitres préparatoires. Sa structure est la suivante. Le premier chapitre introduit certains concepts communs aux deux articles, notamment une description des phénomènes critiques et de la théorie des champs conformes. Le deuxième chapitre aborde brièvement la question des champs logarithmiques, l'évolution de Schramm-Loewner ainsi que l'algorithme de Metropolis-Hastings. Ces sujets sont nécessaires à la lecture de l'article "Geometric Exponents of Dilute Loop Models" au chapitre 3. Le quatrième chapitre présente les outils algébriques utilisés dans le deuxième article, "The idempotents of the TL_n-module \otimes^nC^2 in terms of elements of U_qsl_2", constituant le chapitre 5. La thèse conclut par un résumé des résultats importants et la proposition d'avenues de recherche qui en découlent. / This thesis is concerned with the study of critical phenomena for two-dimensional models on the lattice. Its results are contained in two articles: A first one, devoted to measuring geometric exponents, and a second one to the construction of idempotents for the XXZ spin chain projecting on indecomposable modules of the Temperley-Lieb algebra. Monte Carlo experiments, for a family of loop models in their dilute phase, are presented in the first article. Coined "dilute loop models (DLM)", this family is based upon an O(n) model introduced by Nienhuis (1990). It is defined by two coprime integers p,p' and an anisotropy parameter. In the continuum limit, DLM(p,p') is expected to yield a logarithmic conformal field theory of central charge c(\kappa)=13-6(\kappa+1/\kappa), where the ratio \kappa=p/p' is related to the loop gas fugacity \beta=-2\cos\pi/\kappa. Critical exponents pertaining to valuable geometrical objects, namely the hull, external perimeter and red bonds, were measured. The Metropolis-Hastings algorithm, as well as several methods improving its efficiency, are presented. Despite the extrapolation of curves presenting large slopes, values as close as three to four digits from the theoretical predictions were attained through rigorous statistical analysis. The second article describes the decomposition of the XXZ spin chain Hilbert space \otimes^nC^2 using idempotents. The model of interest (Pasquier & Saleur (1990)) is described by a parameter-dependent Hamiltonian H_{XXZ}(q), q\in C^\times, expressible as a sum of elements of the Temperley-Lieb algebra TL_n(q). The spectrum of H_{XXZ}(q) in the continuum limit is also believed to be related to conformal field theories whose central charge is set by q. Using the quantum Schur-Weyl duality, an expression for the primitive idempotents of End_{TL_n}\otimes^nC^2, involving U_qsl_2 elements, is obtained. These idempotents allow for the explicit construction of the indecomposable TL_n-modules of \otimes^nC^2, all of which are irreducible except when q is a root of unity. This case, and the case where q is generic, are treated separately. Since a wide variety of results and tools are required to tackle the problems stated above, this thesis contains many introductory chapters. Its layout is as follows. The first chapter introduces theoretical concepts common to both articles, in particular an overview of critical phenomena and conformal field theory. Before proceeding to the article entitled \emph{Geometric Exponents of Dilute Loop Models} constituting Chapter 3, the second chapter deals briefly with logarithmic conformal fields, Schramm-Loewner evolution and the Metropolis-Hastings algorithm. The fourth chapter defines some algebraic concepts used in the second article, "The idempotents of the TL_n-module \otimes^nC^2 in terms of elements of U_qsl_2" of Chapter 5. A summary of the main results, as well as paths to unexplored questions, are suggested in a final chapter.