Dans les systèmes de spins quantiques, la frustration et la basse dimensionnalité génèrent des fluctuations quantiques et donnent lieu à des phases exotiques. Cette thèse étudie un modèle d'échelle de spins avec des couplages frustrants le long des montants, motivé par les expériences sur le cuprate BiCu$_2$PO$_6$. Dans un premier temps, on présente une méthode variationnelle originale pour décrire les excitations de basse énergie d'une seule chaîne frustrée. Le diagramme de phase de deux chaînes couplées est ensuite établi à l'aide de méthodes numériques. Le modèle exhibe une transition de phase quantique entre une phase dimérisée est une phase à liens de valence résonnants (RVB). La physique de la phase RVB et en particulier l'apparition de l'incommensurabilité sont étudiées numériquement et par un traitement en champ moyen. On étudie ensuite les effets d'impuretés non-magnétiques sur la courbe d'aimantation et la loi de Curie à basse température. Ces propriétés magnétiques sont tout d'abord discutées à température nulle à partir d'arguments probabilistes. Puis un modèle effectif de basse énergie est dérivé dans la théorie de la réponse linéaire et permet de rendre compte des propriétés magnétiques à température finie. Enfin, on étudie l'effet d'un désordre dans les liens, sur une seule chaîne frustrée. La méthode variationnelle, introduite dans le cas non-désordonné, donne une image à faible désordre de l'instabilité de la phase dimérisée, qui consiste en la formation de domaines d'Imry-Ma délimités par des spinons localisés. Ce résultat est finalement discuté à la lumière de la renormalisation dans l'espace réel à fort désordre. / In quantum spins systems, frustration and low-dimensionality generate quantum fluctuations and give rise to exotic quantum phases. This thesis studies a spin ladder model with frustrating couplings along the legs, motivated by experiments on cuprate BiCu$_2$PO$_6$. First, we present an original variational method to describe the low-energy excitations of a single frustrated chain. Then, the phase diagram of two coupled chains is computed with numerical methods. The model exhibits a quantum phase transition between a dimerized phase and resonating valence bound (RVB) phase. The physics of the RVB phase and in particular the onset of incommensurability are studied numerically and by a mean-field treatment. Afterwards, we study the effects of non-magnetic impurities on the magnetization curve and the Curie law at low temperature. These magnetic properties are first discussed at zero temperature with probability arguments. Then a low-energy effective model is derived within the linear response theory and is used to explain the magnetic properties at finite temperature. Eventually, we study the effect of bonds disorder, on a single frustrated chain. The variational method introduced in the non-disordered case gives a low disorder picture of the dimerized phase instability, which consists in the formation of Imry-Ma domains delimited by localized spinons. This result is finally discussed in the light of the strong disorder real space renormalization.

Propriedades dinâmicas de sistemas quânticos de muitos corpos é um tópico de grande interesse em física da matéria condensada. Estas propriedades nos dão informação sobre a propagação de excitações elementares e de mecanismos de relaxação em sistemas interagentes. Neste contexto, as funções de correlação tem se tornado ainda mais relevantes devido a experimentos em sistemas de átomos frios e íons armadilhados que medem diretamente no domínio temporal os comportamentos assintóticos no tempo. No entanto, até o momento a maioria dos estudos em cadeias de spin quânticas focaram-se em correlações de um único spin. Utilizando a cadeia de spin XX unidimensional, nós estudamos métodos exatos para calcular as funções de correlação das componentes do tensor de dois spins, Tabi,j = SaiSbj. Estes operadores aparecem, por exemplo, como a resposta da seção de choque de espalhamento inelástico de raios X. Baseados no teorema de Wick, nós mostramos que algumas funções de correlação das componentes locais do tensor de dois operadores de spins de sítios vizinhos, na representação de férmions, podem ser escritas como uma combinação de funções de Green de uma única partícula. Utilizamos diagramas de Feynman para organizar esta combinação e calcular as funções de correlação. Em seguida, considerando esses propagadores para tempos longos e grandes distâncias ao longo do cone de luz, encontramos o comportamento dessas funções de correlação como leis de potência oscilatórias que decaem com o tempo e distância. Uma aplicação direta das funções de correlação é para o estudo de quantidades conservadas e não conservadas, uma análise sobre algumas dessas quantidades foi feita. Discutimos também as funções de correlação das componentes do tensor que não são locais na representação fermiônica. Nesse caso os cálculos foram mais desafiadores, mas usamos o fato que funções de correlação dependente do tempo podem ser expressadas em termos dos determinantes de Fredholm. / Dynamical properties of quantum many body systems is a major topic of interest in condensed matter physics. These properties tell us about the propagation of elementary excitation and mechanisms of relaxation in interacting systems. In this context correlation functions have became even more relevant due the experiments in systems of cold atoms and trapped ions that measure real time dependence directly out to relatively long times. However, most studies in quantum spins chains so far have focused on correlations of single spins. Using the one dimensional XX spin chain, we study exact methods to calculate the correlation functions of the components of the tensor operator involving two spins, Tabi,j = SaiSbj. This operator appear, for example, as a response of inelastic x-ray scattering cross section. Based on Wick\'s theorem, we show that some correlation functions of local components of the tensor operator of two pairs of neighbor sites, in the fermion space, can be written as a combination of Greens functions of a single particle. We have used Feynman diagrams to organize this combination and calculate the correlation functions. Then, considering these propagators for long times and large distances along the light cone, we found the behavior of these correlation functions as a oscillatory and power law decay on time. A direct application of correlation functions is to study conserved and non-conserved quantities, and such analysis has been made. We also considered other two-spin operators which are not local in the fermionic representation. In this case the calculation is more challenging, but the time-dependent correlation functions can be expressed in terms of Fredholm determinants.

Propriedades dinâmicas de sistemas quânticos de muitos corpos é um tópico de grande interesse em física da matéria condensada. Estas propriedades nos dão informação sobre a propagação de excitações elementares e de mecanismos de relaxação em sistemas interagentes. Neste contexto, as funções de correlação tem se tornado ainda mais relevantes devido a experimentos em sistemas de átomos frios e íons armadilhados que medem diretamente no domínio temporal os comportamentos assintóticos no tempo. No entanto, até o momento a maioria dos estudos em cadeias de spin quânticas focaram-se em correlações de um único spin. Utilizando a cadeia de spin XX unidimensional, nós estudamos métodos exatos para calcular as funções de correlação das componentes do tensor de dois spins, Tabi,j = SaiSbj. Estes operadores aparecem, por exemplo, como a resposta da seção de choque de espalhamento inelástico de raios X. Baseados no teorema de Wick, nós mostramos que algumas funções de correlação das componentes locais do tensor de dois operadores de spins de sítios vizinhos, na representação de férmions, podem ser escritas como uma combinação de funções de Green de uma única partícula. Utilizamos diagramas de Feynman para organizar esta combinação e calcular as funções de correlação. Em seguida, considerando esses propagadores para tempos longos e grandes distâncias ao longo do cone de luz, encontramos o comportamento dessas funções de correlação como leis de potência oscilatórias que decaem com o tempo e distância. Uma aplicação direta das funções de correlação é para o estudo de quantidades conservadas e não conservadas, uma análise sobre algumas dessas quantidades foi feita. Discutimos também as funções de correlação das componentes do tensor que não são locais na representação fermiônica. Nesse caso os cálculos foram mais desafiadores, mas usamos o fato que funções de correlação dependente do tempo podem ser expressadas em termos dos determinantes de Fredholm. / Dynamical properties of quantum many body systems is a major topic of interest in condensed matter physics. These properties tell us about the propagation of elementary excitation and mechanisms of relaxation in interacting systems. In this context correlation functions have became even more relevant due the experiments in systems of cold atoms and trapped ions that measure real time dependence directly out to relatively long times. However, most studies in quantum spins chains so far have focused on correlations of single spins. Using the one dimensional XX spin chain, we study exact methods to calculate the correlation functions of the components of the tensor operator involving two spins, Tabi,j = SaiSbj. This operator appear, for example, as a response of inelastic x-ray scattering cross section. Based on Wick\'s theorem, we show that some correlation functions of local components of the tensor operator of two pairs of neighbor sites, in the fermion space, can be written as a combination of Greens functions of a single particle. We have used Feynman diagrams to organize this combination and calculate the correlation functions. Then, considering these propagators for long times and large distances along the light cone, we found the behavior of these correlation functions as a oscillatory and power law decay on time. A direct application of correlation functions is to study conserved and non-conserved quantities, and such analysis has been made. We also considered other two-spin operators which are not local in the fermionic representation. In this case the calculation is more challenging, but the time-dependent correlation functions can be expressed in terms of Fredholm determinants.

Ce manuscrit présente l'étude d'une famille d'oxydes de chrome dans laquelle les composés peuvent être synthétisés selon deux phases qui présentent deux topologies différentes : les phases α-ACr₂O₄ (A=Sr, Ca, Ba) et β-CaCr₂O₄. Dans cette famille, les ions chrome portent un spin S=3/2 et forment soit des chaînes triangulaires, soit des plans triangulaires. Notre étude a donc été menée en deux parties : l'étude d'un système magnétique quasi-unidimensionnel et l'étude d'un système quasi-bidimensionnel.La première étude a été consacrée au composé β-CaCr₂O₄. Ce composé présente à basse température un comportement classique caractérisé par un ordre magnétique à longue portée associé à des corrélations 3D et au-dessus de la température d'ordre un comportement quantique caractéristique de chaînes de spins quasi-unidimensionnelles. L'objet de notre étude a donc été de mieux comprendre le mécanisme qui permet la transition d'un régime classique vers un régime quantique dans ce composé. Dans ce contexte nous avons exploré les effets de topologie et de contrainte structurale sur les propriétés magnétiques du composé, à travers l’étude de la série de composés substitués β-CaCr₂-xScxO₄ ainsi qu’une étude sur les effets de la pression. Ce travail a été mis en parallèle avec des calculs théoriques sur les chaînes de spins J₁-J₂.La seconde partie concerne le composé α-SrCr₂O₄. Notamment, l’étude par diffusion des neutrons de ce composé a mis en évidence l’influence de la dimension 2D ainsi que la distorsion du réseau triangulaire sur l’état fondamental du système et son comportement dynamique. / This thesis focuses on the study of a chromium oxide family in which the samples are synthsized into two forms with two different topologies : α-ACr₂O₄ (A=Sr, Ca, Ba) and β-CaCr₂O₄. In these compounds, the chromium ions carry a spin 3/2 and either form triangular spin chains or two dimensional triangular lattices. Our study was hence divided in two parts : the study of a quasi one-dimensional and of a two-dimensional antiferromagnet.The first part was dedicated to the β-CaCr₂O₄ compound. This system exhibits a classical behavior at low temperature associated to a three-dimensional long-range magnetic order and quantum properties typical of a spin chain above the Néel temperature.The aim o four study was therefore to understand the mecanism involved in this classical to quantum physics crossover.In this context, we explored the topological and structural effects on the magnetic properties of this system, through the study of the series of substituted compounds β-CaCr₂-xScxO₄, as well as a study of β-CaCr₂O₄ under pressure. This work was also supported by theoretical calculations on frustrated J₁-J₂ spin chains.The second part of this thesis focused on α-SrCr₂O₄. In particular, neutron scattering measurements on this compound highlighted the effects of the 2D character and the distortion of the triangular lattice on the ground state and dynamical properties in this system.

We describe a reproduction procedure which, given a solution of the gl(m|n) Gaudin Bethe ansatz equation associated to a tensor product of polynomial modules, produces a family P of other solutions called the population. <br>To a population we associate a rational pseudodifferential operator R and a superspace W of rational functions. <br><br>We show that if at least one module is typical then the population P is canonically identified with the set of minimal factorizations of R and with the space of full superflags in W. We conjecture that the singular eigenvectors (up to rescaling) of all gl(m|n) Gaudin Hamiltonians are in a bijective correspondence with certain superspaces of rational functions.<br><br>We establish a duality of the non-periodic Gaudin model associated with superalgebra gl(m|n) and the non-periodic Gaudin model associated with algebra gl(k).<br><br>The Hamiltonians of the Gaudin models are given by expansions of a Berezinian of an (m+n) by (m+n) matrix in the case of gl(m|n) <br>and of a column determinant of a k by k matrix in the case of gl(k). We obtain our results by proving Capelli type identities for both cases and comparing the results.<br><br>We study solutions of the Bethe ansatz equations of the non-homogeneous periodic XXX model associated to super Yangian Y(gl(m|n)).<br>To a solution we associate a rational difference operator D and a superspace of rational functions W. We show that the set of complete factorizations of D is in canonical bijection with the variety of superflags in W and that each generic superflag defines a solution of the Bethe ansatz equation. We also give the analogous statements for the quasi-periodic supersymmetric spin chains.<br>

Ce manuscrit porte sur l’étude de cuprates de basse dimensionnalité, les systèmes à chaînes de spins SrCuO₂ et Sr₂CuO₃. Un des intérêts de ces deux composés est qu’ils présentent des conductions thermiques fortement anisotropes. Celles-ci comportent une contribution magnétique due au transport de la chaleur via les excitations de spinons qui se manifeste uniquement dans la direction des chaînes de spins. Notre étude a pour objectif la mise en évidence des mécanismes qui gouvernent ces propriétés de transport, notamment à travers l’étude des interactions entre les spinons, les phonons et les défauts. Les interactions spinons (phonons)-défauts ont été sondées par l’introduction intentionnelle de dopants (1-2%) non-magnétiques sur le site du cuivre : Mg²⁺, Zn²⁺, Pd²⁺ ou Ni²⁺, ou encore par l’introduction d’éléments possédant des degrés d’oxydation différents sur le site du strontium : La³⁺ ou K⁺. Les composés ont été synthétisés sous leur forme monocristalline par la méthode de fusion de la zone solvante. Des caractérisations structurales, magnétiques et thermiques des composés purs et dopés ont été réalisées. Les spectres d’excitations magnétiques de ces cuprates ont été déterminés par diffusion inélastique de neutrons, spectroscopie RMN et spectroscopie de photoémission résolue en angle afin de révéler l’impact de la substitution. L’étude des spectres de phonons a également été réalisée par diffusion inélastique de neutrons. Les résultats de ces mesures sont corrélés aux propriétés de conduction thermique des composés purs et dopés.. / This manuscript deals with the study of low dimensional cuprates, namely, the spin chains systems SrCuO₂ and Sr₂CuO₃. These two compounds exhibit highly anisotropic thermal conduction properties along the spin-chains direction, where magnetic thermal conduction contributes to the heat transport process via spinon excitations. Our study aims to highlight the mechanisms that govern the heat transport properties, particularly through the study of the scattering channels involving spinon, phonon and defects. The spinon (phonon)–defect scattering was probed by the intentional introduction of nonmagnetic dopants (1-2%) on the copper site, by: Mg²⁺, Zn²⁺, Pd²⁺ or Ni²⁺, or by the introduction of elements carrying different oxidation level on the strontium site, by: La³⁺ or K⁺. Single crystals of the pure and doped materials have been grown by the travelling solvent floating zone method. The structural, magnetic and thermal characterizations of the pure and doped compounds were performed. The magnetic excitation spectra of the compounds were determined by inelastic neutron scattering, NMR spectroscopy, and angle resolved photoemission spectroscopy to reveal the impact of the substitution on the spin dynamics of the doped compounds. The study of phonon spectra has also been performed by inelastic neutron scattering. Results from inelastic neutron scattering have been correlated to the heat transport properties of the pristine and substituted materials.

Ce manuscrit présente l’étude de transitions de phase quantiques dans l’oxyde BaCo2V2O8, un système antiferromagnétique quasi-unidimensionnel constitué de chaînes d’ions cobalt portant un spin effectif S = 1/2 caractérisé par une forte anisotropie de type Ising. Lors de ce travail, nous avons étudié les propriétés statiques et dynamiques de BaCo2V2O8 sous l’effet de différents paramètres physiques.Notre première étude a porté sur l’effet d’un champ magnétique transverse, i.e. appliqué perpendiculairement à l’axe Ising. Il a été proposé que lors de l’application d’un tel champ, un champ magnétique alterné effectif est induit perpendiculairement à l’axe d’anisotropie et au champ uniforme appliqué. La comparaison d'expériences de diffusion (élastique et inélastique) de neutrons et de calculs numériques nous a permis de montrer que ce champ alterné entre en compétition avec l’anisotropie. Ceci aboutit à une transition de phase originale, dite topologique, que l'on peut modéliser par une théorie quantique des champs nommée « modèle de double sine-Gordon » qui décrit la compétition entre deux excitations topologiques duales. Nous avons pu montrer que BaCo2V2O8 sous champ magnétique transverse était la première réalisation d'une telle théorie.La seconde étude était consacrée à BaCo2V2O8 sous champ magnétique longitudinal, i.e. un champ appliqué parallèlement à l’axe Ising. La dynamique de spins a été sondée grâce à la diffusion inélastique de neutrons et nous avons montré qu’au-dessus d’un champ critique de 4 T, celle-ci semble en accord avec le spectre des fluctuations de spin attendu pour un liquide de Tomonaga Luttinger (TLL). De plus, les calculs numériques ont confirmé que, du fait de l’anisotropie de type Ising dans ce système, la majorité du poids spectral du spectre en énergie est porté par les fluctuations de spins de type longitudinales. Ce résultat est la signature d'un comportement quantique sans analogue classique avec des fluctuations de basses énergies essentiellement longitudinales pilotant la physique du système. Enfin, c’est la première fois que la dynamique de spin dans des chaînes de type Ising a pu être sondée dans cette phase TLL.Les deux dernières études sont préliminaires. Le diagramme de phase de BaCo2V2O8 a été sondé par des mesures calorimétriques sous l’application d’une pression hydrostatique et d’un champ magnétique longitudinal. Afin d’obtenir des pressions allant jusqu’à 10 GPa, nous avons utilisé une cellule à enclumes de diamant. Nous avons effectué des mesures de chaleur spécifique qui nous ont permis de sonder l'effet de la pression sur le Hamiltonien de BaCo2V2O8 au travers de son diagramme de phase $(H, P, T)$. Enfin, nous avons étudié l’effet de la substitution des ions magnétiques Co2+ par des impuretés non-magnétiques Mg2+. Les expériences de diffraction neutronique sous champ longitudinal ont montré que la température et le champ critiques diminuent proportionnellement à la concentration en impuretés. La dynamique de spins à champ magnétique nul a aussi été sondée et révèle l’apparition de modes non-dispersifs, provenant possiblement de l’effet de segmentation des chaînes par les impuretés.En conclusion, nos études expérimentales couplées à des calculs numériques nous ont permis de dévoiler une physique extrêmement riche dans ce composé modèle pour l'étude du magnétisme quantique et des transitions de phase quantiques. / This manuscript is devoted to the study of quantum phase transitions in the BaCo2V2O8 oxide, a quasi-one dimensional antiferromagnet consisting of spin chains of cobalt magnetic ions carrying an effective spin S = 1/2 showing a strong Ising-like anisotropy. To achieve this, we have studied BaCo2V2O8 under the effect of different physical parameters.Our first study concerned the effect of a transverse magnetic field, i.e. applied perpendicularly to the Ising axis. It has been shown that when BaCo2V2O8 is subjected to such a field, an effective staggered magnetic field is induced perpendicularly to both the Ising-axis and the uniform applied field. Using neutron scattering experiments (both elastic and inelastic) compared to numerical calculations, we have proved that this staggered field competes with the Ising-like anisotropy. This leads to a very original quantum phase transition. Our system can actually be mapped onto a quantum field theory called “double sine-Gordon model”, describing the competition between two dual topological excitations. We have thus shown that BaCo2V2O8 under a transverse magnetic field is the first experimental realization of such a theory.The second study was devoted to the effect of a longitudinal magnetic field, i.e. a field applied parallel to the Ising-axis. The spin-dynamics have been investigated by means of inelastic neutron scattering experiments and it has been shown that above a critical field of 4 T, it corresponds to the one expected for a Tomonaga Luttinger liquid phase (TLL). Moreover, numerical calculations have shown that, because of the Ising-like anisotropy in this system, the majority of the spectral weight in the energy spectrum is carried by longitudinal spin fluctuations. This result is the signature of a quantum behavior without classical analogous with low energy longitudinal fluctuations driving the physics of the system. Finally, this is the first time that the dispersion spectrum for an Ising-like spin 1/2 chain could be probed in this TLL phase.The last two studies are preliminary work. The phase diagram of BaCo2V2O8 has been probed by calorimetric measurements under pressure and under a longitudinal magnetic field. Pressures up to 10 GPa have been obtained using a diamond anvil cell. We have then performed specific heat measurements allowing us to investigate the effect of pressure on the Hamiltonian of BaCo2V2O8 through its (H, P, T) phase diagram. Finally, we have also started to study the effect of the substitution of magnetic ions Co2+ by non-magnetic impurities Mg2+. The neutron diffraction experiments under a longitudinal magnetic field have shown that the critical temperature and critical field decrease proportionally to the concentration of impurities. The spin-dynamics at zero-field has also been investigated and reveals the appearance of non-dispersive magnetic modes, which possibly comes from the finite size effect of the spin chains segmented by the non-magnetic impurities.In conclusion, our experimental studies associated to numerical calculations allowed us to unveil a very rich physics in this model compound for the study of quantum magnetism and quantum phase transitions.

Le transport thermique dû à des excitations magnétiques dans les composés 1D contenant des chaines et échelles de spins fait actuellement l’objet d’une collaboration fructueuse entre chimistes et physiciens du solide. Le composé à chaines de spins Sr₂CuO₃, qui est un excellent exemple de chaine de type Heisenberg antiferromagnétique de spins S = ½, présente une conductivité thermique anisotrope due à des excitations magnétiques le long des chaines, appelées spinons. Afin de comprendre les mécanismes de diffusion responsables de la conduction thermique magnétique observée, nous avons introduit du désordre dans les chaines en effectuant des substitutions chimiques sur les ions qui composent celles-ci. En appliquant la méthode de croissance dite de “zone solvante” (traveling solvent floating zone method) nous avons élaboré des monocristaux de grande pureté de Sr₂CuO₃ substitués, d’une part par des ions Ca²⁺ (S = 0) qui viennent remplacer des ions Sr²⁺ (S = 0) e, situés en dehors des chaines, créant ainsi un désordre dans les liaisons ; et d’ autre part, des cristaux du même composé, substitué par des ions nickel Ni²⁺ (S = 1) qui prennent cette fois la place des ions Cu²⁺ (S = ½) dans les chaines, introduisant du désordre dans la distribution des sites. Après caractérisation des cristaux par les techniques traditionnelles de diffraction X et chimiques, nous avons fait une étude systématique de leurs propriétés magnétiques et thermiques. Des mesures Raman, de susceptibilité magnétique et de chaleur spécifique ont confirmé l’existence du désordre attendu, et ont suggéré que la substitution par le nickel amenait une rupture des chaines alors que celle par le calcium introduisait des défauts dans les chaines. Par ailleurs, à l’aide de mesures de RMN du ⁶³Cu, nous avons pu observer l’ouverture d’un gap de spins dans le spectre d’excitation des spinons, lié au désordre crée. L’analyse des courbes de conductivité thermique semble indiquer la présence d’une forte interaction spinon-phonon qui limite la conduction thermique à basse température. On notera, en particulier, que pour de forts taux de substitution en calcium, la conductivité thermique croit avec la température, mettant ainsi en évidence l’origine magnétique, associée aux spinons, de la conduction thermique. / Heat transport due to magnetic excitations in strongly correlated electronic materials containing low dimensional spin structures such as chains and ladders is an emerging field of research due to a fruitful collaboration between condensed matter physicists and solid state chemists. The spin chain compound Sr₂CuO₃, which is known to be the best realization of the S = ½ Heisenberg antiferromagnetic model, exhibits anisotropic thermal conductivity due to magnetic excitations, called spinons, of the spin chains. In order to understand its magnetic thermal conduction and the nature of scattering mechanisms involved, we have induced disorder into the chain structure by carrying out chemical substitution. Employing the sophisticated traveling solvent floating zone method we have grown high purity single crystals of Ca-substituted Sr₂CuO₃, in which Ca²⁺ (S = 0) ions substitute Sr²⁺ (S = 0) ions situated off the spin chains in the lattice creating a bond disorder, and Ni-substituted Sr₂CuO₃ in which Ni²⁺ (S = 1) ions substitute Cu²⁺ (S = ½) ions in the spin chains creating a site disorder. After characterizing the crystals using standard techniques, we have performed a systematic study of their magnetic and thermal properties. Raman spectra, magnetic susceptibility and specific heat measurements performed on these crystals confirmed the introduction of the anticipated disorder, suggesting that the substituted Ni atoms lead to breaking of the spin chains, and the Ca atoms act as effective in-chain defects. We observed the phenomenon of opening of a spingap in the otherwise gapless spinon excitation spectrum due to the induced disorder using ⁶³Cu NMR measurements. Our analysis of the thermal conductivity measurements points towards the presence of a strong spinon-phonon scattering that limits heat conduction at low temperatures. Remarkably, substituting high concentrations of Ca, we observed that thermal conductivity increases linearly with temperature, revealing the intrinsic magnetic thermal conduction due to spinons.

The quantum Hall (QH) system---cold electrons in two dimensions in a perpendicular magnetic field---is a striking example of a system where unexpected phenomena emerge at low energies. The low-energy physics of this system is effectively one-dimensional due to the magnetic field. We identify an exactly solvable limit of this interacting many-body problem, and provide strong evidence that its solutions are adiabatically connected to the observed QH states in a similar manner as the free electron gas is related to real interacting fermions in a metal according to Landau's Fermi liquid theory. The solvable limit corresponds to the electron gas on a thin torus. Here the ground states are gapped periodic crystals and the fractionally charged excitations appear as domain walls between degenerate ground states. The fractal structure of the abelian Haldane-Halperin hierarchy is manifest for generic two-body interactions. By minimizing a local k+1-body interaction we obtain a representation of the non-abelian Read-Rezayi states, where the domain wall patterns encode the fusion rules of the underlying conformal field theory. We provide extensive analytical and numerical evidence that the Laughlin/Jain states are continuously connected to the exact solutions. For more general hierarchical states we exploit the intriguing connection to conformal field theory and construct wave functions that coincide with the exact ones in the solvable limit. If correct, this construction implies the adiabatic continuation of the pertinent states. We provide some numerical support for this scenario at the recently observed fraction 4/11. Non-QH phases are separated from the thin torus by a phase transition. At half-filling, this leads to a Luttinger liquid of neutral dipoles which provides an explicit microscopic example of how weakly interacting quasiparticles in a reduced (zero) magnetic field emerge at low energies. We argue that this is also smoothly connected to the bulk state.

In dieser Dissertation untersuchen wir unterschiedliche Aspekte im Zusammenhang mit Korrelationsfunktionen in der Fischnetz-Theorie. Zunächst betrachten wir einen der einfachsten Korrelatoren der Fischnetz Theorie, das konforme Box-Integral, in Minkowski Signatur. Während dieses Integral in Euklidischer Signatur eine konforme Symmetrie aufweist, wird diese Symmetrie in Minkowski-Raumzeit subtil gebrochen. Wir beschreiben die Brechung der konformen Symmetrie quantitativ, indem wir die funktionale Form des Box-Integrals in allen kinematischen Regionen untersuchen. Ausserdem untersuchen wir das Ausmass zu dem das Box integral durch seine Yangian-Symmetrie festgelegt ist. Als nächstes widmen wir uns den Basso-Dixon-Graphen, die ebenfalls konforme Vier-Punkt-Integrale sind und Verallgemeinerungen des Box-Integrals zu höheren Schleifenordnungen darstellen. Wir leiten die Yangian-Ward-Identitäten ab, die diese Klasse von Integralen erfüllen. Die Ward-Identitäten sind einhomogene Erweiterungen der partiellen Differentialgleichungen, die im homogenen Fall durch Appell-Hypergeometrische Funktionen gelöst werden. Die Ward-Identitäten können natürlicherweise auf eine Ein-Parameter-Familie von D-dimensionalen Integralen erweitert werden, die Korrelatoren in der verallgemeinerten Fischnetz-Theorie von Kazakov und Olivucci darstellen. Schliesslich untersuchen wir den Dilatationsoperator in einem Drei-Skalar-Sektor der Fischnetztheorie, der auch als Eklektisches Modell bezeichnet wird. In diesem Sektor der Dilatationsoperator nimmt nicht--diagonalisierbare Form an. Das führt dazu, dass die Zwei-Punkt-Korrelationsfunktionen eine logarithmische Abhängigkeit von der Raumzeitseparierung der Operatoren annimmt. Unter Zuhilfenahme von kombinatorischen Argumenten führen wir eine generierende Funktion ein, die das Jordan-Block-Spektrum eines verwandten Modells, der hypereklektischen Spinkette, vollständig charakterisiert. / We study various aspects of correlation functions in fishnet theory. We begin with the study of the simplest correlator in theory theory, represented by the conformal box integral, in Minkowski space. While this integral is conformally invariant in Euclidean space, this symmetry is subtly broken in Minkowski space. We quantify the extent to which conformal symmetry is broken by analysing the functional form of the box in each kinematic region. We propose a new method to calculate the box integral directly in Minkowski space, by introducing a family of configurations with two points at infinity. Furthermore, we investigate the extent to which the box integral is constrained by Yangian symmetry. We constrain the functional form of the box integral in all kinematic regions up to twelve undetermined constants, which we fix by three separate analytic continuations from the Euclidean region. Next, we study the Basso-Dixon graphs, which represent higher-loop versions of the box integral. We derive and study Yangian Ward identities for this class of integrals. These take the form of inhomogeneous extensions of the partial differential equations defining the Appell hypergeometric functions. The Ward identities naturally generalise to a one-parameter family of D dimensional integrals representing correlators in a generalised fishnet theory. Finally, we study the dilatation operator in a particular three scalar sector of the fishnet theory, which has been dubbed the eclectic model. This dilatation operator is non-diagonalisable in this sector. This leads to logarithmic spacetime dependence in the corresponding two-point functions. Using combinatorial arguments, we introduce a generating function which fully characterises the Jordan block spectrum of a related model: the hypereclectic spin chain. This function is found by purely combinatorial means and can be expressed in terms of the q-binomial coefficient.