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We characterise the higher order Riesz transforms on the Heisenberg group and also show that they satisfy dimension-free bounds under some assumptions on the multipliers. We also prove the boundedness of the higher order Riesz transforms associated to the Hermite operator. Using transference theorems, we deduce boundedness theorems for Riesz transforms on the reduced Heisenberg group and hence also for the Riesz transforms associated to special Hermite and Laguerre expansions. Next we study the Riesz transforms associated to the Grushin operator G = - Δ - |x|2@t2 on Rn+1. We prove that both the first order and higher order Riesz transforms are bounded on Lp(Rn+1): We also prove that norms of the first order Riesz transforms are independent of the dimension n.

In this PhD thesis, the physics of vacancies in two-dimensional ordered Heisenberg antiferromagnets is investigated. We use semi-classical methods to study the influence of a single vacancy in long-range ordered states, with a focus on non-collinear order. Here, on a classical level, a magnetic distortion is created as the spins readjust in response to the vacancy. We use the non-collinear $120^\\circ$ state on the frustrated triangular lattice as an example, where we determine the impurity contributions to the magnetization and susceptibility. An important discovery is the vacancy moment not being quantized due to non-universal partial screening. The resulting effective moment $m_0 \\ll S$ can be observed as a fractional prefactor to an impurity-induced Curie response $m_0^2/(3k_BT)$ at finite temperature. This is in sharp contrast to collinearly ordered states. Here the moment is always quantized to the bulk spin value, $m_0=S$. Furthermore, we present a detailed analysis of the vacancy-induced distortion cloud. Due to Goldstone modes, it decays algebraically as $r^{-3}$ with distance $r$ to the vacancy. Using leading-order $1/S$-expansion, we determine the quantum corrections to both size and direction of the distorted magnetic moments. Secondly, we study the same problem in the presence of an external magnetic field $h$, both for the square and triangular lattice. For the triangular lattice we use a biquadratic exchange term $K$ to stabilize a unique ground state from a degenerate manifold. The finite-field vacancy moment $m(h)$ is generated by field-dependent screening clouds, as different non-collinear bulk states evolve with increasing field. These distortion clouds decay exponentially on a magnetic length scale $l_h\\propto 1/h$. Most importantly, we find that the magnetic-field linear-response limit $h \\rightarrow 0^+$ is generically singular for $SU(2)$ ordered local-moment antiferromagnets, as the vacancy moment in zero field differs fundamentally from even an infinitesimal but finite field, $m(h \\rightarrow 0^+)\\neq m_0$. Moreover, a part of the screening cloud itself becomes universally singular. Particularly for spin-flop states, this leads to a semi-classical version of perfect screening. We present general arguments to support these claims, as well as microscopic calculations. Another remarkable result is an impurity-induced quantum phase transition for overcompensated vacancies in the $M=1/3$ plateau phase on the triangular lattice with $K<0$. We close our analysis with a discussion about important limits for finite vacancy concentrations, as well as a possible experimental verification of our predictions.

Ce mémoire examine, de manière critique, les réflexions apportées par Wemer Heisenberg quant au rôle du langage dans la poursuite de la compréhension de la nature par la physique. Après avoir cerné brièvement les influences philosophiques ayant marqué la pensée de Heisenberg, nous examinons comment, selon ce dernier, la compréhension que nous donne du monde la physique se démarque des simples descriptions de la nature ainsi que des autres types de compréhensions humaines. Le rôle du langage dans l'atteinte de cette compréhension est ensuite examiné. Il en ressort que les trois langages utilisés par la physique (le langage ordinaire, le langage scientifique et le langage symbolique des mathématiques) ont chacun une fonction différente à jouer dans le développement de cette science, fonction déterminée par leurs différentes façons de signifier ainsi que le besoin de la physique de faire à la fois appel aux mathématiques et à l'expérience.

Doctor of Philosophy / Department of Mathematics / Louis Crane / We exhibit a static, cylindrically symmetric, exact solution to the Euler-Heisenberg field equations (EHFE) and prove that its effective geometry contains (optical) black holes. It is conjectured that there are also soliton solutions to the EHFE which contain black hole geometries.

We determine the dynamical structure factors of two gapped correlated electron systems, namely the Ising model in a strong transverse field and the two-leg spin-1/2 Heisenberg ladder in the limit of strong rung coupling. We consider the low-temperature limit, employing a variety of analytical and numerical techniques. The coherent modes of single-particle excitations, which are delta functions at zero temperature, are shown to broaden asymmetrically in energy with increasing temperature. Firstly, we apply a low-temperature “resummation” inspired by the Dyson equation to a linked-cluster expansion of the two-leg Heisenberg ladder. We include matrix elements to second order in the interaction between states containing up to two particles. A low-frequency response similar to the “Villain mode” is also observed. Next, we apply a cumulant expansion technique to the transverse field Ising model. We resolve the issue of negative spectral weight caused by double pole in the leading self-energy diagram by including a resummation of terms obtained from the six-point function, demonstrating that the perturbation series in 2n-spin correlation functions can be extended to higher orders. The result generalises to higher dimensions and the analytic calculation is compared to a numerical Pade approximant. We outline the extension of this method to the strong coupling ladder. Finally, we compare the previous results to numerical data obtained by full diagonalisation of finite chains and numerical evaluation of the Pfaffian, a method specific to the transverse field Ising chain. The latter method is used for a phenomenological study of the asymmetric broadening as well as an evaluation of fitting functions for the broadened lineshapes.

Nesta dissertação estudamos sistemas de spins em redes de baixa dimensionalidade e em temperatura nula, analisando suas transições de fases quânticas. Mais precisamente, estu- damos as propriedades do estado fundamental e as possíveis transições de fase do modelo de Heisenberg quântico antiferromagnético de spin-1/2, com interações entre os primeiros e segundos vizinhos, em diversas redes, e em particular na rede triangular, que é o foco de nosso estudo. Para a obtenção do estado fundamental aproximado, usamos um método variacional em que a rede é particionada num conjunto de plaquetas de sítios. O estado fundamental é escrito como um produto tensorial dos estados das plaquetas. Para a rede triangular, escolhemos um triângulo como uma plaqueta. Quatro fases foram encontra- das: a fase antiferromagnética de Néel, a colinear, a fase de Néel modificada e aquela que denominamos de ligação covalente ressonante. Obtivemos as energias e as magnetizações de subrede em função da razão entre as interações de primeiros e segundos vizinhos. En- tre as fases de Néel e a colinear, podemos observar a fase de ligação covalente ressonante caracterizada como um singleto quanto ao spin de cada plaqueta. / In this thesis we study spin systems in low-dimensional lattices at zero temperature, analyzing their quantum phase transitions. More precisely, we study the properties of the ground state and the possible phase transitions in the antiferromagnetic spin-1/2 quan- tum Heisenberg model with interaction between the first and second neighbors, in several lattices, and in particular in the triangular lattice, which is the focus of our study. To obtain the approximate ground state, we use a variational method in which the lattice is partitioned into a set of plates of sites. The ground state is written as a tensor product of the states of plates. For the triangular lattice, we choose a triangle as a plate. Four phases were found: the antiferromagnetic Néel phase, the collinear, the modified Néel phase and that we call resonating valence bond. We obtained the energy and the magnetization as a function of the ratio of the interactions between the first and second neighbor sites. Between the Néel and collinear phases, we can observe the spin resonating valence bond phase, characterized as a singlet with respect to the spin of each plate.

In this thesis we study actions of countable groups on measure spaces underthe assumption that the dynamics are non-singular, with particular reference topointwise ergodic theorems and their relationship to the critical dimensions ofthe action.

Nesta dissertação estudamos sistemas de spins em redes de baixa dimensionalidade e em temperatura nula, analisando suas transições de fases quânticas. Mais precisamente, estu- damos as propriedades do estado fundamental e as possíveis transições de fase do modelo de Heisenberg quântico antiferromagnético de spin-1/2, com interações entre os primeiros e segundos vizinhos, em diversas redes, e em particular na rede triangular, que é o foco de nosso estudo. Para a obtenção do estado fundamental aproximado, usamos um método variacional em que a rede é particionada num conjunto de plaquetas de sítios. O estado fundamental é escrito como um produto tensorial dos estados das plaquetas. Para a rede triangular, escolhemos um triângulo como uma plaqueta. Quatro fases foram encontra- das: a fase antiferromagnética de Néel, a colinear, a fase de Néel modificada e aquela que denominamos de ligação covalente ressonante. Obtivemos as energias e as magnetizações de subrede em função da razão entre as interações de primeiros e segundos vizinhos. En- tre as fases de Néel e a colinear, podemos observar a fase de ligação covalente ressonante caracterizada como um singleto quanto ao spin de cada plaqueta. / In this thesis we study spin systems in low-dimensional lattices at zero temperature, analyzing their quantum phase transitions. More precisely, we study the properties of the ground state and the possible phase transitions in the antiferromagnetic spin-1/2 quan- tum Heisenberg model with interaction between the first and second neighbors, in several lattices, and in particular in the triangular lattice, which is the focus of our study. To obtain the approximate ground state, we use a variational method in which the lattice is partitioned into a set of plates of sites. The ground state is written as a tensor product of the states of plates. For the triangular lattice, we choose a triangle as a plate. Four phases were found: the antiferromagnetic Néel phase, the collinear, the modified Néel phase and that we call resonating valence bond. We obtained the energy and the magnetization as a function of the ratio of the interactions between the first and second neighbor sites. Between the Néel and collinear phases, we can observe the spin resonating valence bond phase, characterized as a singlet with respect to the spin of each plate.

On considère le groupe de Heisenberg $\He_n=\R^{2n+1}$ avec la distance de Carnot-Carathéodory $d_c$ et la mesure de Lebegue $\Lg^{2n+1}$. Dans le premier chapitre, dans le cadre du problème du voyageur de commerce géométrique de $\Hei$, on construit une courbe de longueur finie qui ne vérifie pas le critère de Ferrari, Franchi et Pajot au sujet des ensembles contenus dans une courbe rectifiable. On montre aussi une inégalité sur le déterminant jacobien des applications de contraction sur un point qui suivent les géodésiques. Cette inégalité est essentiellement équivalente à la Propriété de Contraction de Mesure $MCP(0,2n+3)$. Grâce à cette proprété on répond positivement au Chapitre 2 à une question d'Ambrosio et Rigot à propos du transport de mesure dans $\He_n$ (travail en commun avec Figalli). Il s'avère en effet que les mesures traversées par une géodésique de l'espace de Wasserstein sont absolument continues dès qu'une extrémité de la géodésique l'est. Au Chapitre 3 on démontre que la Courbure-Dimension $CD(K,N)$ définie par transport de mesure n'est pas vérifiée pour $\He_n$ et que cela vaut quels que soient les paramètres $K\in\R$ et $N\in[1,+\infty]$. On discute aussi d'autres propriétés de courbures dans le cas du groupe de Heisenberg. Le Chapitre 4 est dédié à la correspondance entre l'équation de la chaleur sous-elliptique et le flot de gradient de l'entropie de Bolzmann dans l'espace de Wassertein.