This dissertation is devoted to theoretical studies of strongly interacting one-dimensional and quasi one-dimensional electron systems. The properties of one-dimensional electron systems can be studied within the bosonization technique, which presents fermions as collective bosonic density excitations. The power of this approach is the ability to treat electron-electron interaction exactly in the low-energy limit. The approach predicts the failure of Fermi liquid and an absence of long-range order in one-dimensions. The low-energy description of one-dimensional interacting systems is called the Tomonaga-Luttinger liquid theory. For example, the edges of quantum Hall systems are one-dimensional and described by a chiral Tomonaga-Luttinger liquid. Another example is a quantum spin Hall system with helical edge states, which are also described by a Tomonaga-Luttinger liquid. In our first work, a study of magnetized edge states of quantum spin-Hall system is presented. A magnetic field dependent signature of such edges is obtained, which can be verified in a Coulomb drag experiment. The second part of the dissertation is devoted to quasi-one dimensional antiferromagnetic lattices. A spatially anisotropic lattice antiferromagnet can be viewed as an array of one dimensional spin chains coupled in a way to match the lattice symmetry. This allows to use the non-Abelian bosonization technique to describe the low-energy physics of spin chains and study the inter-chain interactions perturbatively. The work presented in the dissertation studies the effect of Dzyaloshinskii-Moriya interaction on the magnetic phase diagram of the spatially anisotropic kagome antiferromagnet. We predict a Dzyaloshinskii-Moriya interaction driven phase transition from Neel to Neel+dimer state. In the third work, a novel model of the fractional quantum Hall effect is given. Wave functions of two-dimensional electrons in strong and quantizing magnetic field are essentially one-dimensional. That invites one to use the one-dimensional phenomenological bosonization to describe the density fluctuations of the two-dimensional interacting electrons in magnetic field. Remarkably, the constructed trial bosonized fermion operator describing the electron states with a fixed Landau gauge momentum is effectively two-dimensional. / text

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Strongly correlated systems on geometrically frustrated lattices can stabilize a large number of interesting phases that includes a wide array of novel Mott insulators in both bosonic and electronic systems. Charge fluctuations in a Mott insulator are suppressed due to strong mutual interaction among the particles. The presence of frustration is of particular importance as the physics it offers is often rich, unexpectedly complicated, and continues to raise many open questions. The thesis elucidates some of these issues on a kagome lattice where strong interactions among the particles in the Mott phase impose non-trivial local constraints depending on the filling fraction on the lattice. These Mott insulators, in addition to featuring unusual magnetic and/or charge ordering, can also harbor topologically ordered states of quantum matter, e.g., resonating valence bond liquids realized in certain quantum dimer models on non-bipartite lattices. The dimer models can be regarded as low-energy effective theories for different types of bosonic models in the strong-coupling limit. Exploring this connection is a central theme of this thesis with the aim of realizing novel strongly correlated ground states. Past studies of these models have revealed the existence of various ordered and disordered phases with distinct signatures. Among these low-energy phases, the presence of a stable topological liquid at a particular point, known as Rokhsar-Kivelson point, in the phase diagram is notable. The classical versions of the dimer model are also known to have garnered a vast interest in various fields ranging from problems of pure mathematical origin to ones in physical chemistry as well as statistical physics. Pioneered by Kasteleyn, several analytical works came forward to exactly calculate the partition function of the problem from which other physical observables can be derived. Classical numerical methods are extensively applied to these models to verify the analytical predictions. We introduce a new classical algorithm here to compute the correlation functions of a classical dimer model on a square (bipartite) and a triangular (non-bipartite) lattice based on a tensor network construction. The method, called tensor network renormalization group, turns out to be a powerful tool for simulating short-ranged gapped systems as inferred from our results benchmarked against the classical Monte-Carlo technique and compared with past analytical studies. One should note that the quantum dimer model at the Rokhsar-Kivelson point can also be described as an infinite temperature canonical ensemble of classical dimers because of the particular structure of the ground state which is an equal weight superposition in the configuration manifold. The geometry of the lattice plays a pivotal role in deciding the nature of the phases that arise in the dimer models. Many physical properties of the dimer liquid phase can be extracted in the simple classical setting which certainly allows for a deep understanding of the classical models to be developed. The liquid phase is gapped on non-bipartite lattices and gapless on bipartite lattices, which is reflected in the decay of correlation functions with spatial distances. In general on non-bipartite lattices, the topological nature of the dimer liquid is characterized by a Z2 topological order which survives even when the model is perturbed away from the Rokhsar-Kivelson point. Stability of this liquid phase not only depends on the lattice geometries but notably on dimer concentrations also. In this context, we focus on a particular variant of the dimer model on a triangular lattice which is known as the quantum fully packed loop model. The model is composed of nonintersecting closed loops made of dimers and governed by the same Hamiltonian as the quantum dimer model. The loop model provides an effective low-energy description of a strongly correlated bosonic system at 1/3 filling on the kagome lattice. The corresponding Bose-Hubbard Hamiltonian consists of nearest-neighbor hopping and all possible repulsive interactions within a hexagonal plaquette. Conspicuous features of the zero-temperature phase diagram for this model include (i) presence of a stable Z2 liquid even without any Rokhsar-Kivelson potential term (in distinction to the standard quantum dimer model), and (ii) an unconventional phase transition from the liquid phase to a novel crystalline phase that has nematic order (dubbed lattice nematic). For a deeper understanding of the physics, a mapping to an Ising gauge theory is presented. The gauge theoretic description provides a useful way to predict the nature of the quantum phase transition to lie in the O(3) universality class. Finally a fermionic model at the same 1/3 filling is considered in which the ground state exhibits a number of exotic local orderings resulting from the spin-charge interplay of electrons. The Hamiltonian comprises nearest-neighbor hopping, strong on-site Coulomb interaction, and repulsive interaction terms only between nearest-neighbors. In the strong correlation limit, this fermionic problem maps to a two-color fully packed loop model – a model in which the loop segments carry an additional quantum number as color on a honeycomb lattice. The effective theory is governed by coherent three-particle ring exchanges and nearest-neighbor antiferromagnetic spin exchanges. The competition between these two leads to a phase diagram composed of a novel plaquette ordered state (known as the plaquette phase) that undergoes phase transition to a new kind of charge ordered state which we call a short loop phase. From our numerical analysis, we conclude that the plaquette phase features an unusual antiferromagnetic order with gapless spin excitations while the charge-ordered state is subjugated by spin fluctuations of localized electrons arranged in small hexagonal loops on the kagome lattice.

Cette thèse a porté sur les principes de guidage, les propriétés linéaires et les outils de conception autour des fibres à cristal photonique à coeur creux (HC-PCF) à couplage inhibé (IC). Le guidage IC a été démontré comme une manifestation photonique de Q-BiC (état quasi lié dans un continuum) en étudiant des profils asymétriques et dépendants en polarisation dit Fano présentant une bande passante spectrale de 30 GHz. En utilisant le concept de IC, nous reportons la caractérisation linéaire de fibres IC HC-PCF supérieures à l’état de l’art. Par une optimisation de la forme du coeur, une fibre Kagome IC HC-PCF a démontré des pertes très faibles de 8,5 dB/km à 1030 nm associées à une bande passante à 3 dB de 225 nm. Une autre conception avec des entretoises de silice amincies à 300 nm a permis d’atteindre des pertes de 30 dB/km à 780 nm avec une bande de transmission fondamentale record décalée à 670 nm et capable de couvrir toutes les gammes spectrales du Ti:Sa, Yb et Er. Nous avons également travaillé sur la conception et la fabrication de IC HC-PCF présentant une gaine dont la structure est un réseau unique de tubes fins isolés. Une de ces fibres a permis de démontrer une transmission jusqu'à 220 nm avec des pertes records de 7,7 dB/km à ~ 750 nm, tandis qu’une seconde réalisation s’est traduit par une bande fondamentale de plus d’une octave allant de 600 à 1200 nm avec des pertes de 10-20 dB/km. Finalement, cette dernière fibre a été étudiée plus en détail pour déterminer les sources à l’origine des pertes due à la rugosité de surface présente à l’interface du contour du coeur. / This thesis reported on guiding principles, linear properties and conceptual design tools of inhibited coupling (IC) guiding hollow-core photonic crystal fibers (HC-PCF). IC guidance was proved as photonic manifestation of Q-BiC (quasi bound-state-in-a-continuum) by investigating asymmetric and polarization dependent Fano profiles with bandwidth of 30 GHz in high resolution transmission spectra. By using IC design concept, we reported on linear characterization of state-of-the-art IC HC-PCFs. Based on core shaping optimization, a Kagome IC HC-PCF demonstrated ultra-low loss down to 8.5 km/km at 1030 nm associated with a 225 nm wide 3-dB bandwidth. Another Kagome design with thinner silica struts of 300 nm exhibited lowest loss of 30 dB/km at 780 nm along with record level fundamental bandwidth spreading down to 670 nm and able to cover the entire Ti:Sa, Yb and Er laser spectral ranges. We also reported on design and fabrication of single-ring tubular lattice IC HC-PCFs. One of these fibers demonstrated transmission down to 220 nm with a record transmission loss of 7.7 dB/km at ~750 nm, while the second one exhibited ultra-broad fundamental band with loss range of 10-20 dB/km over one octave spanning from 600 to 1200 nm. Finally, the second tubular fiber was further investigated for fundamental loss sources due to surface roughness around its core-contour.

Strongly correlated systems on geometrically frustrated lattices can stabilize a large number of interesting phases that includes a wide array of novel Mott insulators in both bosonic and electronic systems. Charge fluctuations in a Mott insulator are suppressed due to strong mutual interaction among the particles. The presence of frustration is of particular importance as the physics it offers is often rich, unexpectedly complicated, and continues to raise many open questions. The thesis elucidates some of these issues on a kagome lattice where strong interactions among the particles in the Mott phase impose non-trivial local constraints depending on the filling fraction on the lattice. These Mott insulators, in addition to featuring unusual magnetic and/or charge ordering, can also harbor topologically ordered states of quantum matter, e.g., resonating valence bond liquids realized in certain quantum dimer models on non-bipartite lattices. The dimer models can be regarded as low-energy effective theories for different types of bosonic models in the strong-coupling limit. Exploring this connection is a central theme of this thesis with the aim of realizing novel strongly correlated ground states. Past studies of these models have revealed the existence of various ordered and disordered phases with distinct signatures. Among these low-energy phases, the presence of a stable topological liquid at a particular point, known as Rokhsar-Kivelson point, in the phase diagram is notable. The classical versions of the dimer model are also known to have garnered a vast interest in various fields ranging from problems of pure mathematical origin to ones in physical chemistry as well as statistical physics. Pioneered by Kasteleyn, several analytical works came forward to exactly calculate the partition function of the problem from which other physical observables can be derived. Classical numerical methods are extensively applied to these models to verify the analytical predictions. We introduce a new classical algorithm here to compute the correlation functions of a classical dimer model on a square (bipartite) and a triangular (non-bipartite) lattice based on a tensor network construction. The method, called tensor network renormalization group, turns out to be a powerful tool for simulating short-ranged gapped systems as inferred from our results benchmarked against the classical Monte-Carlo technique and compared with past analytical studies. One should note that the quantum dimer model at the Rokhsar-Kivelson point can also be described as an infinite temperature canonical ensemble of classical dimers because of the particular structure of the ground state which is an equal weight superposition in the configuration manifold. The geometry of the lattice plays a pivotal role in deciding the nature of the phases that arise in the dimer models. Many physical properties of the dimer liquid phase can be extracted in the simple classical setting which certainly allows for a deep understanding of the classical models to be developed. The liquid phase is gapped on non-bipartite lattices and gapless on bipartite lattices, which is reflected in the decay of correlation functions with spatial distances. In general on non-bipartite lattices, the topological nature of the dimer liquid is characterized by a Z2 topological order which survives even when the model is perturbed away from the Rokhsar-Kivelson point. Stability of this liquid phase not only depends on the lattice geometries but notably on dimer concentrations also. In this context, we focus on a particular variant of the dimer model on a triangular lattice which is known as the quantum fully packed loop model. The model is composed of nonintersecting closed loops made of dimers and governed by the same Hamiltonian as the quantum dimer model. The loop model provides an effective low-energy description of a strongly correlated bosonic system at 1/3 filling on the kagome lattice. The corresponding Bose-Hubbard Hamiltonian consists of nearest-neighbor hopping and all possible repulsive interactions within a hexagonal plaquette. Conspicuous features of the zero-temperature phase diagram for this model include (i) presence of a stable Z2 liquid even without any Rokhsar-Kivelson potential term (in distinction to the standard quantum dimer model), and (ii) an unconventional phase transition from the liquid phase to a novel crystalline phase that has nematic order (dubbed lattice nematic). For a deeper understanding of the physics, a mapping to an Ising gauge theory is presented. The gauge theoretic description provides a useful way to predict the nature of the quantum phase transition to lie in the O(3) universality class. Finally a fermionic model at the same 1/3 filling is considered in which the ground state exhibits a number of exotic local orderings resulting from the spin-charge interplay of electrons. The Hamiltonian comprises nearest-neighbor hopping, strong on-site Coulomb interaction, and repulsive interaction terms only between nearest-neighbors. In the strong correlation limit, this fermionic problem maps to a two-color fully packed loop model – a model in which the loop segments carry an additional quantum number as color on a honeycomb lattice. The effective theory is governed by coherent three-particle ring exchanges and nearest-neighbor antiferromagnetic spin exchanges. The competition between these two leads to a phase diagram composed of a novel plaquette ordered state (known as the plaquette phase) that undergoes phase transition to a new kind of charge ordered state which we call a short loop phase. From our numerical analysis, we conclude that the plaquette phase features an unusual antiferromagnetic order with gapless spin excitations while the charge-ordered state is subjugated by spin fluctuations of localized electrons arranged in small hexagonal loops on the kagome lattice.

Topological insulators are electronic materials that behave like an ordinary insulator in their bulk but have robust conducting states on their edge. Besides, in some materials the band structure presents completely flat bands, a special feature leading to strong interactions effects. In this thesis we present a study of the edge states of three particular two-dimensional models presenting flat bands: the honeycomb-Kagome, the $\alpha$--graphyne and a ligand decorated honeycomb-Kagome lattice models. We extend earlier work done on these lattice models by focusing on the topological nature of the edge states involving flat bands. We start by giving a review of the band structure theory and the tight-binding approximation. We then present several main topics in two-dimensional topological insulators such as the notion of topological invariants, the Kane-Mele model and the bulk-edge correspondence. Using these theoretical concepts we study the band structure of these lattices firstly without taking into account the spin and spin-orbit interations. We finally add these interactions to get their bulk band structures as well as the edge states. We observe how these spin-orbit interactions relieve degeneracies and allow for the emergence of edge states of topological nature. Since the lattices studied have an arrangement based on the honeycomb-Kagome lattice, two-dimensional materials having the structures of these lattices can be designed assembling metal ions and organic ligands. Therefore the results obtained could be used as a first hint to create new two-dimensional materials presenting topological properties. / Topologiska isolatorer är elektroniska material som uppför sig som en vanlig isolator i sin bulk men har robusta ledande stater på kanten. Dessutom presenterar bandstrukturen i vissa material helt platta band, en speciell egenskap som leder till starka interaktionseffekter. I denna avhandling presenterar vi en studie av kanttillstånden för tre speciella tvådimensionella modeller som presenterar platta band: bikakan-Kagome, $\alpha$-grafynen och en liganddekorerad honungskaka-Kagome modeller. Vi utökar tidigare arbete med dessa gittermodeller genom att fokusera på den topologiska karaktären hos kanttillstånd som innefattar platta band. Vi börjar med att ge en genomgång av bandstruktursteorin och den tätt bindande approximationen. Vi presenterar sedan flera huvudämnen i tvådimensionella topologiska isolatorer såsom begreppet topologiska invarianter, Kane-Mele modellen och bulk-kant korrespondensen. Med hjälp av dessa teoretiska begrepp studerar vi bandstrukturen för dessa gitter först utan att ta hänsyn till spinnen och spinnsorbital interaktioner. Vi lägger sedan till dessa interaktioner för att få sina bulkbandstrukturer såväl som kanttillstånden. Vi observerar hur dessa spinnsorbital interaktioner lindrar degenerationer och möjliggör uppkomsten av kanttillstånd av topologisk naturen. Eftersom de undersökta gitterna har ett arrangemang baserat på honungskaka-Kagome gitteren, kan tvådimensionella material med strukturerna hos dessa gitter utformas genom att montera metalljoner och organiska ligander. Därför kan de erhållna resultaten användas som en första ledtråd för att skapa nya tvådimensionella material med topologiska egenskaper.

Strong electron correlation lies at the root of many quantum collective phenomena observed in solids, including high Tc superconductivity. Theoretically, the problem of many interacting electrons is difficult to treat, however, and a microscopic understanding of strongly correlated systems remains one of the foremost challenges in modern physics. A particularly clean realisation of this general problem is found in magnetic systems, where theory and experiment are both well developed and complementary. The role of the chemist in this endeavour is to provide model experimental systems to both inspire new developments in theory and to confirm existing predictions. This thesis aims to demonstrate aspects of both synthesis and physical characterisation of such model systems, with particular emphasis on materials which exhibit unusual quantum ground states due to a combination of reduced dimensionality, low spin, and geometric frustration. Four materials are considered: The first among these is a new material, KTi(SO4)2·(H2O), which was prepared using a hydrothermal route, and characterised by magnetic susceptibility, specific heat, and high field magnetisation measurements. Fitting exact diagonalisation and series expansion results to these data imply that KTi(SO4)2·(H2O)is a long-sought experimental realization of the S = 1/2 Heisenberg frustrated (J1 − J2) chain model in the dimerised regime of the phase diagram. The anhydrous analogue of KTi(SO4)2·(H2O), KTi(SO4)2, was also investigated, and found by magnetic neutron scattering to exemplify the S = 1/2 Heisenberg anisotropic triangular lattice model in the 1D chain limit. The final two materials discussed are the naturally occurring minerals volborthite and herbertsmithite, both thought to realise the S = 1/2 Heisenberg kagome antiferromagnet model. Diffuse and inelastic magnetic neutron scattering experiments, however, indicate that the kagome physics are partially destroyed by defects in the former and lattice distortion in the latter.

Dans les composés antiferromagnétiques Heisenberg à géométrie kagomé, la frustration des interactions est à l'origine d'un état liquide de spins à T=0. Un état RVB, originalement proposé par Anderson dans les réseaux triangulaires puis dans les cuprates, décrit probablement ce fondamental, dont l'état singulet est dégénéré exponentiellement dans un faible gap singulet-triplet. Peu de composés expérimentaux se rapprochent aujourd'hui du système idéal et l'existence de perturbations ou d'anisotropie lève souvent la dégénérescence du fondamental lorsque T->0. Une étude par RMN, MuSR et SQUID, a été réalisée dans les composés à géométrie de bicouches kagomé de spin 3/2, Ba2Sn2ZnGa10-7pCr7pO22 et SrCr9pGa12-9pO19, considérés comme les archétypes d'un hamiltonien de spins purement Heisenberg sur un réseau kagomé. Outre l'absence caractéristique de transition jusqu'à une température Tg~2 K bien plus basse que la température de Curie-Weiss thetaCW~250 K, des propriétés physiques semblables dans ces deux systèmes sont mesurées malgré des défauts radicalement différent. Les propriétés intrinsèques de cette géométrie en sont déduites, à savoir: - une décroissance de la susceptibilité en dessous de 45 K (RMN) interprétée par de très faibles longueurs de corrélations magnétiques, au moins pour T>10 K, malgré les fortes interactions antiferromagnétiques (thetaCW~250 K). Ce maximum de la susceptibilité reste compatible avec l'existence d'un gap de spin; - l'existence de fluctuations quantiques pour T>30 mK, beaucoup plus bas que Tg (MuSR); - la corrélation entre l'apparition de ces fluctuations et celle d'un état de type verre de spin à Tg. Un modèle phénoménologique décrivant la relaxation des muons est présenté, pour la première fois dans ces systèmes, et suggère la stabilisation d'un état de type RVB en dessous de Tg.

Cette these est consacree a l'etude de quelques aspects theoriques de systemes geometriquement frustres. Dans certains reseaux (pyrochlore,...) ou les interactions antiferromagnetiques entre spins (S=1/2) plus proches voisins sont fortement frustrees l'etat fondamental ainsi que les premiers etats excites sont des singulets (S=0). Par ailleurs, les correlations spin-spin sont a courte portee dans ces liquides de spins. En prenant en compte ces deux caracteristiques, ainsi que la geometrie du reseau pyrochlore, on derive un hamiltonien effectif pour decrire ce systeme a basse temperature. En resolvant cet hamiltonien a l'approximation du champ moyen, on obtient la structure singulet de l'etat fondamental. Le reseau kagome a ete l'objet de nombreuses etudes qui montrent que l'hamiltonien de Heisenberg conduit a un etat desordonné liquide de spins sur ce reseau. Ceci est en desaccord avec l'ordre magnetique observe dans certaines realisations experimentales du reseau kagome (de la famille des jarosites). On montre que les interactions Dzyaloshinsky-Moriya sont autorisees par la geometrie du reseau kagome (et des jarosites), on en precise la structure et une derivation microscopique de ces interactions est faite. Les proprietes magnetiques qui en decoulent sont etudiees dans l'approximation du champ moyen et par simulations Monte Carlo. Les fluctuations quantiques sont sondees par un developpement en ondes de spins. On montre que les interactions de Dzyaloshinsky-Moriya expliquent la mise en ordre de certains composes jarosites. Le cas du reseau pyrochlore est egalement considere : on explicite les interactions Dzyaloshinsky-Moriya autorisees par la geometrie de ce reseau et les structures magnetiques qui en decoulent. Cette interaction pourrait expliquer l'ordre magnetique qui apparai t dans certains composes pyrochlores.

Les lasers à fibre de haute puissance constituent depuis une dizaine d’année un outil pertinent pour un nombre croissant d’applications. Dans le cadre d’un contrat CIFRE entre la société Eolite Systems et le laboratoire Xlim (UMR 7252 du CNRS et de l’Université de Limoges), mon projet de thèse a consisté à développer les briques technologiques de futures sources lasers / High-power fiber lasers adress an increasing number of applications since ten years. In the frame of a CIFRE contract between the company Eolite Systems and Xlim (joint laboratory between CNRS and the University of Limoges), the goal of this PhD project was to develop the technological blocs to achieve all-fibre high-power lasers emiting out of the conventional spectral band covered by existing lasers.Modal instabilities in large mode area (LMA) fibers are currently the main limitation of the fiber lasers power scaling. We have experimentally demonstrated the relevance of inner cladding aperiodic structures to efficiently delocalize higher order modes outside the gain region. A systematic study of passive fibers based on such structures has shown the single propagation of the fundamental mode over a wide wavelength range from 1 to 2 µm for dimension of core up to 85 µm. This effective mode delocalization even extends up to a core dimension of 140 µm at a 2 µm wavelength.The combination of high power picosecond fiber laser with an average power of 22.7 W and a hydrogen-filled inhibited coupling Kagome fiber allowed us to generate two Raman combs over five frequency octaves from 321 nm to 12.5 µm. These two combs are controlled by the laser pump polarization and generated an average power of 10.1 W displayed over 70 laser lines for circular pump polarization and 8.6 W over 30 lines for linear polarization. Some laser lines within these combs have been generated for the first time from high-power fiber source in the mid-infrared range. We have also demonstrated the generation of high-power line by optimizing the first vibrational Stokes at 1.8 µm with an average power of 9.3 W and a quantum efficiency of the frequency conversion stage close to 80%.