En este trabajo estudiamos el efecto de una modulación de intercambio aperiódica sobre el modelo del collar de Kondo. La relación de dispersión para las excitaciones del sistema es obtenida empleando una representación para los espines localizados y de conducción en términos de los operadores locales singlete y triplete. Esto es realizado en el marco de una aproximación Gaussiana, a temperatura cero y finita y para una dimensión arbitraria d. Los resultados permiten estudiar dos fases: una paramagnética a temperatura cero y otra antiferromagnética a temperatura finita, aunque con muy bajos valores (cerca del cero absoluto). En el primer caso se estudia la dependencia del gap de las energías del espín con la modulación de intercambio aperiódica, mientras que en el segundo caso se determina la línea de transición de Neel también como función de la modulación de intercambio aperiódica. / -- In this work we have studied the aperiodic exchange modulation on the Kondo necklace model. The dispersion relation for the excitations of the system is obtained using a representation for the localized and conduction electrons in terms of local Kondo singlet and triplet operators. This result was obtained using a Gaussian approximation, at finite and zero temperature and arbitrary dimension, d. Our results allowed the study of two phases: the paramagnetic at zero tempe-rature and the antiferromagnetic at low finite temperature. In the first case, we studied the spin gap dependence on the aperiodic exchange modulation, whereas, in the second one we obtained the critical Neel line also as a function of the aperiodic exchange modulation. / Tesis

Nas últimas duas décadas a teoria dos sistemas eletrônicos correlacionados teve enorme progresso, que sustentou o paralelo desenvolvimento da pesquisa experimental dos sistemas de férmions pesados. Dada a complexidade do problema proposto pelas correlações fortes, diversas técnicas complementares de cálculo foram desenvolvidas no período. O presente plano se propõe a explorar uma extensão de uma das mais antigas, a técnica do grupo de renormalização numérico (GRN), tratando perturbativamente o modelo de Kondo para uma impureza magnética em um hospedeiro metálico. É bem conhecido que a expansão perturbativa de propriedades físicas, como a susceptibilidade, em termos do acoplamento de troca diverge logaritmicamente próxima da temperatura de Kondo. A abordagem do GRN para isto considera a transformação discreta, T[HN] = HN+1, onde {HN} é uma seqüência de Hamiltonianos. Neste trabalho, para regularizar a expansão da susceptibilidade, usamos um procedimento alternativo considerando a transformação contínua análoga, T&#948z[HN(z)] = HN(z+&#948z), onde z é um parâmetro arbitrário que generaliza a discretização logarítmica do GRN. Ao contrário do procedimento de Wilson, nós esperamos que este novo procedimento possa ser mais facilmente aplicável a Hamiltonianos mais complexos, complementando a diagonalização numérica. / In the last two decades the theory of electronic correlated systems has had an enormous progress, which has sustained the parallel development of the experimental research in heavy fermion systems. Given the complexity imposed by the strong correlations, several techniques appeared. The present work explores an extension of one of the oldest, the Numerical Renormalization Group (NRG), treating perturbatively the Kondo model for a magnetic impurity in a metallic host. It is well known that perturbative expansion of physical properties, like susceptibility, in terms of the exchange coupling diverges logarithmically near the Kondo temperature. The NRG approach for this consider the discrete transformation, T[HN] = HN+1, where {HN}, is a sequence of Hamiltonians. In this work we use an alternative procedure to regularize the expansion, using an analogous continuum transformation T&#948z[HN(z)] = HN(z+&#948z), where z is an arbitrary parameter which generalizes the NRG logarithmic discretization. Unlike Wilson\'s procedure, we hope this new one can be easily applicable to more complex Hamiltonians, complementing the numerical diagonalization.

Made available in DSpace on 2015-04-22T22:07:26Z (GMT). No. of bitstreams: 1 FRANCISCO_ DINOLA.pdf: 881552 bytes, checksum: 816e42b2af8f05ee4d548310017c668e (MD5) Previous issue date: 2007-03-26 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / In this work we have calculated the magnetic susceptibility χ, the specific heat C and a electrical resistivity ρ for a system composed by a impurity inserted in a metal, described by a Fermi level resonance, utilizing a density of states obtained from X-ray Photoelectron Spectroscopy spectral density in the Kondo regime using the Anderson model obtained by Frota and Oliveira using the renormalization group technique in T = 0, in sequence generalized by Chattopadyay and Jarrel for T 6= 0, adjusting the results obtained by Quantum Monte Carlo technique, adding to the resonant level width Γ a factor γ(T) = 4.52T . Our results are in qualitative agreement with exact results obtained to Kondo model. / Neste trabalho calcularemos a susceptibilidade magnética χ, o calor específico C e a resistividade elétrica ρ de um sistema constituído por uma impureza embebida em um metal, representado por uma ressonância no nível de Fermi, usando como densidade de estados a densidade espectral obtida da espectroscopia fotoeletronica de raio-X para o modelo de Anderson no limite do regime Kondo, obtida por Frota e Oliveira usando o método de Grupo de Renormalização para T = 0, posteriormente generalizado por Chattopadhyay e Jarrel para T 6= 0, ajustando o resultado obtido pela técnica de Monte Carlo Quântico, acrescentando à largura Γ do nível ressonante o fator γ(T) = 4.52T . Os resultados obtidos estão em acordo qualitativo com os resultados exatos obtidos para o modelo de Kondo.

Nas últimas duas décadas a teoria dos sistemas eletrônicos correlacionados teve enorme progresso, que sustentou o paralelo desenvolvimento da pesquisa experimental dos sistemas de férmions pesados. Dada a complexidade do problema proposto pelas correlações fortes, diversas técnicas complementares de cálculo foram desenvolvidas no período. O presente plano se propõe a explorar uma extensão de uma das mais antigas, a técnica do grupo de renormalização numérico (GRN), tratando perturbativamente o modelo de Kondo para uma impureza magnética em um hospedeiro metálico. É bem conhecido que a expansão perturbativa de propriedades físicas, como a susceptibilidade, em termos do acoplamento de troca diverge logaritmicamente próxima da temperatura de Kondo. A abordagem do GRN para isto considera a transformação discreta, T[HN] = HN+1, onde {HN} é uma seqüência de Hamiltonianos. Neste trabalho, para regularizar a expansão da susceptibilidade, usamos um procedimento alternativo considerando a transformação contínua análoga, T&#948z[HN(z)] = HN(z+&#948z), onde z é um parâmetro arbitrário que generaliza a discretização logarítmica do GRN. Ao contrário do procedimento de Wilson, nós esperamos que este novo procedimento possa ser mais facilmente aplicável a Hamiltonianos mais complexos, complementando a diagonalização numérica. / In the last two decades the theory of electronic correlated systems has had an enormous progress, which has sustained the parallel development of the experimental research in heavy fermion systems. Given the complexity imposed by the strong correlations, several techniques appeared. The present work explores an extension of one of the oldest, the Numerical Renormalization Group (NRG), treating perturbatively the Kondo model for a magnetic impurity in a metallic host. It is well known that perturbative expansion of physical properties, like susceptibility, in terms of the exchange coupling diverges logarithmically near the Kondo temperature. The NRG approach for this consider the discrete transformation, T[HN] = HN+1, where {HN}, is a sequence of Hamiltonians. In this work we use an alternative procedure to regularize the expansion, using an analogous continuum transformation T&#948z[HN(z)] = HN(z+&#948z), where z is an arbitrary parameter which generalizes the NRG logarithmic discretization. Unlike Wilson\'s procedure, we hope this new one can be easily applicable to more complex Hamiltonians, complementing the numerical diagonalization.

Dans cette thèse, nous étudions d'un point de vue théorique différents aspects de la matière topologique. Ces systèmes présentent des propriétés résistantes aux éventuelles perturbations grâce à une topologie non-triviale de leur structure de bandes. En particulier, des excitations exotiques, par exemple des fermions de Majorana, peuvent apparaitre à leurs bords.L'entropie d'intrication, ainsi que le spectre d'intrication ont été fondamentaux dans l'étude théorique de ces systèmes, et plus généralement des phases libres. Il est cependant difficile de les mesurer expérimentalement. L'étude des fluctuations de charge bipartites a été proposée afin de remédier à ce problème, et celles-ci permettent une mesure faible de l'intrication, en particulier pour des modèles unidimensionnels libres. Nous généralisons les précédents travaux sur les Liquides de Luttinger à des familles génériques de supraconducteurs et isolants topologiques en une et deux dimensions, systèmes dans lesquels la charge observée n'est plus conservée. Nous montrons que les transitions de phases topologiques sont caractérisées par certains coefficients universels dans les fluctuations et les fonctions de corrélations. Les systèmes bidimensionnels que nous étudions présentent des cônes de Dirac, et ces coefficients dépendent de leur enroulement. Cela nous permet de caractériser la topologie de ces points critiques. Dans tous les cas, les fluctuations suivent une loi de volume, qui a un comportement non-analytique aux transition de phase.Dans un second temps, nous nous intéressons aux systèmes en interactions. Nous montrons tout d'abord que certaines des signatures des transitions topologiques survivent en leur présence, dans les supraconducteurs topologiques. Nous étudions ensuite le diagramme de phase de deux fils supraconducteurs couplés par une interaction Coulombienne. Celle-ci mène à la création de phases exotiques grâce à la compétition avec la supraconductivité non-conventionnelle. Nous montrons en particulier l'apparition de phases de Mott brisant spontanément la symétrie de renversement du temps et présentant des courant orbitaux non-triviaux, ainsi que celle d'une phase de fermions libres, qui est l'extension de deux chaînes de Majorana critiques en interaction.Enfin, nous nous intéressons aux effets de la présence de fermions de Majorana sur le transport électronique. Nous étudions un îlot supraconducteur où plusieurs de ces fermions existent. Ce système pourrait être l'un des composants élémentaires d'un éventuel ordinateur quantique. Les fermions de Majorana changent les statistiques d'échange des porteurs de charges, ce qui se traduit par une fractionnalisation de la conductance. Celle-ci se révèle très robuste face aux anisotropies et autres perturbations. Nous étendons les études précédentes au cas où le nombre d'électrons dans la boîte peut fluctuer, et montrons l'équivalence de ce problème avec le modèle Kondo à plusieurs canaux. Nous réinterprétons alors ce modèle en terme du déplacement d'une particule dans un réseau fictif dissipatif. / In this thesis, we study theoretically different aspects of topological systems. These models present resilient properties due to a non-trivial topology of their band structures, and in particular exotic edge excitations such as Majorana fermions.Entanglement entropy and entanglement spectrum have been fundamental to the study of these systems and of gapless systems in general, but are difficult to measure experimentally. Bipartite charge fluctuations were proposed as a weak measurement of this entanglement, in particular for one-dimensional gapless phases. We extend previous results on standard Luttinger Liquids to generic families of one- and two-dimensional non-interacting topological systems. Through exact computations, we show that their critical points are characterized by universal coefficients that reveal the topological aspect of the transitions. In two dimensions, the Dirac cones give quantized contributions to the fluctuations and various correlation functions. These contributions depend on their winding numbers, allowing for a precise determination of the topological structure of the gapless points. A volume law is also present and linked to the Quantum Fisher information, with characteristic non-analyticities at the phase transitions.In a second time, we include interactions and show that some of these signatures are preserved in topological superconductors even in their presence. Through analytical (bosonization, renormalization group) and numerical (exact diagonalization and DMRG) methods, we study the phase diagram of two Coulomb-coupled topological superconducting wires. We are interested in their behavior when the interactions are strong enough to break the topological protection: the interplay between unconventional superconductivity and interactions leads to exotic phases. We show the appearance of phases spontaneously breaking the time-reversal symmetry, with non-trivial orbital currents, and of an unusual gapless phase that is the extension of two critical interacting Majorana modes.Finally, we are interested in electronic transport mediated by Majorana fermions. We study a floating superconducting island carrying several such impurities. This device is thought to be a potential building block for a quantum computer. The Majorana fermions affect the statistics of the charge carriers, which leads to very resilient fractionalized transport. We extend previous studies to the charge degenerate case, where the total number of fermions in the island is not fixed, and map it to the well-known Multi-Channel Kondo model at large interaction. We reinterpret this standard model in terms of a particle moving in a highly dimensional, dissipative lattice.

We apply an antiferromagnetic symmetry breaking implementation of the dynamical cluster approximation (DCA) to investigate the two-dimensional hole-doped Kondo lattice model (KLM) with hopping $t$ and coupling $J$. The DCA is an approximation at the level of the self-energy. Short range correlations on a small cluster, which is self-consistently embedded in the remaining bath electrons of the system, are handled exactly whereas longer ranged spacial correlations are incorporated on a mean-field level. The dynamics of the system, however, are retained in full. The strong temporal nature of correlations in the KLM make the model particularly suitable to investigation with the DCA. Our precise DCA calculations of single particle spectral functions compare well with exact lattice QMC results at the particle-hole symmetric point. However, our DCA version, combined with a QMC cluster solver, also allows simulations away from particle-hole symmetry and has enabled us to map out the magnetic phase diagram of the model as a function of doping and coupling $J/t$. At half-filling, our results show that the linear behaviour of the quasi-particle gap at small values of $J/t$ is a direct consequence of particle-hole symmetry, which leads to nesting of the Fermi surface. Breaking the symmetry, by inclusion of a diagonal hopping term, results in a greatly reduced gap which appears to follow a Kondo scale. Upon doping, the magnetic phase observed at half-filling survives and ultimately gives way to a paramagnetic phase. Across this magnetic order-disorder transition, we track the topology of the Fermi surface. The phase diagram is composed of three distinct regions: Paramagnetic with {\it large} Fermi surface, in which the magnetic moments are included in the Luttinger sum rule, lightly antiferromagnetic with large Fermi surface topology, and strongly antiferromagnetic with {\it small} Fermi surface, where the magnetic moments drop out of the Luttinger volume. We draw on a mean-field Hamiltonian with order parameters for both magnetisation and Kondo screening as a tool for interpretation of our DCA results. Initial results for fixed coupling and doping but varying temperature are also presented, where the aim is look for signals of the energy scales in the system: the Kondo temperature $T_{K}$ for initial Kondo screening of the magnetic moments, the Neel temperature $T_{N}$ for antiferromagnetic ordering, a possible $T^{*}$ at which a reordering of the Fermi surface is observed, and finally, the formation of the coherent heavy fermion state at $T_{coh}$. / Wir setzen eine Implementierung der dynamischen Cluster Näherung (DCA) mit gebrochener Symmetrie ein um das zweidimensionale lochdotierte Kondo Gitter Model (KLM) mit dem Hüpfmatrixelement $t$ und der Kopplung $J$ zu untersuchen. Die DCA beruht auf einer Näherung der Selbstenergie. Kurzreichweitige Korrelationen auf einem kleinen Cluster, der selbstkonsistent in ein Bad der übrigen Systemelektronen eingebettet ist, werden exakt behandelt, während langreichweitige Korrelationen auf Mean-Field Basis berücksichtigt werden. Dabei wird jedoch die Dynamik des Systems voll beibehalten. Auf Grund starker dynamischer Korrelationen zeigt sich das KLM als besonders geeignet für Untersuchungen im Rahmen der DCA. Präzise Berechnungen der Einteilchen Spektralfunktion geben gute Übereinstimmung mit exakten Gitter-QMC Resultaten am Teilchen-Loch symmetrischen Punkt. Unsere DCA Version, kombiniert mit einem QMC Cluster Solver, erlaubt es, Simulationen fern vom Teilchen-Loch symmetrischen Punkt durchzuführen und hat es uns ermöglicht das magnetische Phasendiagram des Models als Funktion der Dotierung und der Kopplung $J/t$ abzutasten. Bei halber Füllung zeigen unsere Resultate, dass das lineare Verhalten der Quasiteilchenlücke bei kleinem $J/t$ direkt aus der vorliegenden Teilchen-Loch Symmetrie, die ihrerseits zu Nesting führt, hervorgeht. Brechung dieser Symmetrie durch das Einführen eines diagonalen Hüpfmatrixelements, hat eine an die Kondo Skala gekoppelte, stark reduzierte Quasiteilchenlücke zur Folge. Im dotiertem System setzt sich die bei Halbfüllung beobachtete magnetische Phase fort bis sie letztendlich der paramagnetischen Phase weicht. Wir verfolgen die Entwicklung der Topologie der Fermifläche beim Durchstoßen dieses magnetischen Übergangs vom Ordnungs- zum Unordnungregime. Das Phasendiagram unterteilt sich in drei verschiedenen Regionen: Den Paramagnetischen Bereich mit {\it großer} Fermifläche, in dem die magnetische Momente zum Luttinger Volumen beitragen, den schwachen Antiferromagneten, mit großer Fermiflächetopologie, und den starken Antiferromagneten mit {\it kleiner} Fermifläche, bei dem die magnetischen Momente nicht am Luttinger Volumen beteiligt sind. Wir beziehen uns zur weiteren Interpretation unserer DCA Resultate auf einen Mean-Field Hamiltonian mit Ordnungsparametern sowohl für die Magnetisierung als auch für die Kondo-Abschirmung. Erste Resultate bei fester Kopplung und Dotierung, jedoch bei unterschiedlichen Temperaturen, zwecks der Ermittlung der verschiedene Energieskalen des Systems, werden dargestellt. Wir suchen Signale der Kondo Temperatur $T_{K}$ bei der die Kondo-Abschirmung der magnetische Momente einsetzt, der Neel Temperatur $T_{N}$ der antiferromagnetischem Ordnung, das eventuelle Auftreten einer durch $T^{*}$ gekennzeichnete Änderung der Fermiflächen Topologie, und letztendlich die Ausbildung eines kohärenten schwerfermionischen Zustandes bei $T_{coh}$.

Thesis (Ph.D.)--University of Central Florida, 2009. / Adviser: Michael N. Leuenberger. Includes bibliographical references (p. 104-115).

Stuttgart, Univ., Diss., 2000.

Stuttgart, Univ., Diss., 2001.

Konstanz, Univ., Diss., 2004.