Au cours de cette thèse, nous avons étudié le comportement critique de chaînes de spins quantiques en présence de couplages désordonnés ou répartis de manière apériodique. Il est bien établi que le comportement critique des chaînes de spins quantiques d’Ising et de Potts est gouverné par le même point fixe de désordre infini. Nous avons implémenté́ une version numérique de la technique de renormalisation de désordre infini (SDRG) afin de tester cette prédiction. Dans un second temps, nous avons étudié la chaîne quantique d’Ashkin-Teller désordonnée par renormalisation de la matrice densité́ (DMRG). Nous confirmons le diagramme de phase précédemment proposé en déterminant la position des pics du temps d’autocorrélation intégré des corrélations spin-spin et polarisation-polarisation ainsi que ceux des fluctuations de l’aimantation et de la polarisation. Enfin, l’existence d’une double phase de Griffiths est confirmée par une étude détaillée de la décroissance des fonctions d’autocorrélation en dehors des lignes critiques. Comme attendu, l’exposant dynamique diverge à l’approche de ces lignes. Dans le cas apériodique, nous avons étudié les chaînes quantiques d’Ising et de Potts. En utilisant la méthode SDRG, nous avons confirmé les résultats connus pour la chaîne d’Ising et proposé des estimations de la dimension d’échelle magnétique. Dans le cas du modèle de Potts à q états, nous avons estimé l’exposant magnétique et observé qu’il était indépendant du nombre d’états q pour toutes les séquences dont l’exposant de divagation est nul. Toutefois, nous montrons que l’exposant dynamique est fini et augmente avec le nombre d’états q. En revanche, pour la séquence de Rudin-Shapiro, les résultats sont compatibles avec un point fixe de désordre infini et donc un exposant dynamique infini. / In the present thesis, the critical and off-critical behaviors of quantum spin chains in presence of a random or an aperiodic perturbation of the couplings is studied. The critical behavior of the Ising and Potts random quantum chains is known to be governed by the same Infinite-Disorder Fixed Point. We have implemented a numerical version of the Strong-Disorder Renormalization Group (SDRG) to test this prediction. We then studied the quantum random Ashkin-Teller chain by Density Matrix Renormalization Group. The phase diagram, previously obtained by SDRG, is confirmed by estimating the location of the peaks of the integrated autocorrelation times of both the spin-spin and polarization-polarization autocorrelation functions and of the disorder fluctuations of magnetization and polarization. Finally, the existence of a double-Griffiths phase is shown by a detailed study of the decay of the off-critical autocorrelation functions. As expected, a divergence of the dynamical exponent is observed along the two transition lines. In the aperiodic case, we studied both the Ising and Potts quantum chains. Using numerical SDRG, we confirmed the known analytical results for the Ising chains and proposed a new estimate of the magnetic scaling dimension.For the quantum q-state Potts chain, we estimated the magnetic scaling dimension for various aperiodic sequences and showed that it is independent of q for all sequences with a vanishing wandering exponent. However, we observed that the dynamical exponent is finite and increases with the number of states q. In contrast, for the Rudin-Shapiro sequence, the results are compatible with an Infinite-Disorder Fixed Point with a diverging dynamical exponent, equipe de renormalization

Na aproximação de potencial local (L\\downarrow1) a transformação de grupo de renormalização para o modelo O(N) de Heisenberg hierárquico é descrita por uma equação a derivadas parciais (EDP). Neste trabalho investigamos, na criticalidade (sistema à temperatura inversa crítica), a somabilidade da série de potências em 1/N que formalmente satisfaz essa EDP. / In the local potential approximation (L\\downarrow1) the renormalization group transformation for the hierarchical O(N) Heisenberg model is described by a partial differential equation (PDE). In this work we investigate, at criticality (system at inverse critical temperature), the summability of the formal power series in 1/N which formally satisfies that PDE.

In this thesis, we investigate the entanglement and magnetic properties of frustrated spin systems and correlated electronic properties of conjugated carbon systems. In chapter 1, we present different approaches to solve the time-independent, nonrelativistic Schr¨odinger equation for a many-body system. We start with the full non-relativistic Hamiltonian of a multi nuclear system to describe the Born - Oppenheimer approximation which allows the study of electronic Hamiltonian which treats nuclear positions parametrically. We then also describe ab initio techniques such as the Hartree-Fock Method and density functional theories. We then introduce model Hamiltonians for strongly correlated systems such as the Hubbard, Pariser-Parr-Pople and Heisenberg models, and show how they result from the noninteracting one-band tight-binding model. In chapter 2, we discuss various numerical techniques like the exact diagonalization methods and density matrix renormalization group (DMRG) method. We also discuss quantum entanglement and the success of DMRG which can be attributed to the area law of entanglement entropy. In chapter 3, we study here different regions in phase diagrams of the spin-1/2, spin-1 and spin-3/2 one-dimensional antiferromagnetic Heisenberg systems with nearest-neighbor (J1) and next-nearest-neighbor (J2) interactions and dimerization (d ). Frustration arises for specific relative signs of the interactions J1 and J2. In particular, we analyze the behavior of the bipartite entanglement entropy and fidelity at the gapless to gapped phase transitions and across the lines separating different phases in the J2−d plane. All the calculations in this work are based on exact diagonalizations of finite systems. In chapter 4, we study Heisenberg spin-1/2 and spin-1 chains with alternating ferromagnetic (JF 1 ) and antiferromagnetic (JA 1 ) nearest-neighbor interactions and a ferromagnetic next-nearest-neighbor interaction (JF 2 ). In this model frustration is present due to non-zero JF 2 . The model with site spin s behaves like a Haldane spin chain with site spin 2s in the limit of vanishing JF 2 and large JF 1 /JA 1 . We show that the exact ground state of the model can be found along a line in the parameter space. For fixed JF 1 , the phase diagram in the space of JA 1 −JF 2 is determined using numerical techniques complemented by analytical calculations. A number of quantities, including the structure factor, energy gap, entanglement entropy and zero temperature magnetization, are studied to understand the complete phase diagram. An interesting and potentially important feature of this model is that it can exhibit a macroscopic magnetization jump in the presence of a magnetic field; we study this using an effective Hamiltonian. In chapter 5, we study correlated electronic properties of zigzag and armchair fused naphthalenes and polyperylene systems in the presence of long-range electronelectron interactions. We find that the ground state of zigzag fused naphthalene system is a higher spin state, while the ground state of armchair fused naphthalene is a singlet. The spin gap of polyperylene is unusually small and the ground state is a singlet. Our calculations of optical gap and two-photon gap suggest that polyperylene should exhibit fluorescence. From the charge gap calculation, we predict that in zigzag fused naphthalene and polyperylene systems, excitons are weakly binding. Peierls type of distortion is negligible in zigzag fused naphthalene and polyperylene systems, however, in armchair fused naphthalene system, interior bonds have tendency to distort in low-lying excited states. In chapter 6, we study the ground state spin of the Heisenberg spin-1/2 nearestneighboring antiferromagnetic exchange models of systems with fused odd member rings. In particular, we compute the ground state spin of fused three and five membered rings as well as fused five membered rings. In the thermodynamic limit, the ground state of the fused three and five membered system is a higher spin state, while fused five membered system shows a singlet ground state, for all system sizes.

Dans la première partie de cette thèse, je présenterai le 5D MSSM qui est un modèle super symétrique avec une dimension supplémentaire. (Five Dimensional Minimal Supersymmetric Standard Model). Apres compactification sur l'orbifold S1/Z2, le calcul des équations du groupe de renormalisation (RGE) à une boucle montre un changement dans l'évolution des paramètres phénoménologiques. Dès que l'énergie E = 1/R est atteinte, les états de Kaluza- Klein interviennent et donnent des contributions importantes. Plusieurs possibilités pour les champs de matière sont discutés : ils peuvent se propager dans le "bulk" ou ils sont localisés sur la "brane". Je présenterai d'une part l'évolution des équations de Yukawa dans le secteur des quarks ainsi que les paramètres de la matrice CKM, d'autre part, les effets de ce modèle sur le secteur des neutrinos notamment les masses, les angles de mélange, les phases de Majorana et de Dirac. Dans la deuxième partie, je parlerai du modèle AMSB et ses extensions (MM-AMSB et HCAMSB). Ces modèles sont des scenarios de brisure assez bien motivés en super symétrie. En calculant des observables issues de la physique des particules puis en imposant des contraintes de cosmologie standard et alternative sur ces scénarios, j'ai déterminé les régions qui respectent les contraintes de la matière noire et les limites de la physique des saveurs. Je reprendrai ensuite l'analyse de ces modèles en utilisant de nouvelles limites pour les observables. La nouvelle analyse est faite en ajoutant les mesures récentes sur la masse du Higgs et les rapports de branchement pour plusieurs canaux de désintégrations / In the first part of this thesis, we review the Universal Extra-Dimensional Model compactified on a S1/Z2 orbifold, and the renormalisation group evolution of quark and lepton masses, mixing angles and phases both in the UED extension of the Standard Model and of the Minimal Supersymmetric Standard Model (the five-dimensional MSSM). We consider two typical scenarios: all matter fields propagating in the bulk, and matter fields constrained on the brane. The two possibilities give rise to quite different behaviours. For the quark sector we study the Yukawa couplings and various flavor observables and for the neutrino sector, we study the evolution of neutrino masses, mixing angles and phases. The analysis is performed in the two cases for different values of tan β and different radii of compactification. The resulting renormalization group evolution equations in these scenarios are compared with the existing results in the literature, together with their implications. In the second part, we present a simulation study about anomaly mediated supersymmetry breaking and its extensions. Anomaly mediation is a popular and well motivated supersymmetry breaking scenario. Different possible detailed realisations of this set-up are studied and actively searched for at colliders. Apart from limits coming from flavour, low energy physics and direct collider searches, these models are usually constrained by the requirement of reproducing the observations on dark matter density in the universe. We reanalyse these bounds and in particular we focus on the dark matter bounds both considering the standard cosmological model and alternative cosmological scenarios. We briefly discuss the implications for phenomenology and in particular at the Large Hadron Collider. After that we update our analysis by using new limits from observables and adding recent Higgs boson measurements for the mass and signal strengths in different decay channels

Dans cette thèse, nous nous intéressons à la renormalisation de Connes et Kreimer dans le contexe des algèbres de Hopf de graphes de Feynman spécifiés. Nous construisons une structure d'algèbre de Hopf $\mathcal{H}_\mathcal{T}$ sur l'espace des graphes de Feynman spécifié d'une théorie quantique des champs $\mathcal{T}$. Nous définissons encore un dédoublement $\wt\mathcal{D}_\mathcal{T}$ de la bigèbre de graphes de Feynman spécifiés, un produit de convolution \divideontimes et un groupe de caractères de cette algèbre de Hopf à valeurs dans une algèbre commutative qui prend en compte la dépendance en les moments extérieurs. Nous mettons en place alors la renormalisation décrite par A. Connes et D. Kreimer et la décomposition de Birkhoff pour deux schémas de renormalisation : le schéma minimal de renormalisation et le schéma de développement de Taylor. Nous rappelons la définition des intégrales de Feynman associées à un graphe. Nous montrons que ces intégrales sont holomorphes en une variable complexe D dans le cas des fonctions de Schwartz, et qu'elles s'étendent en une fonction méromorphe dans le cas des fonctions de types Feynman. Nous pouvons alors déterminer les parties finies de ces intégrales en utilisant l'algorithme BPHZ après avoir appliqué la procédure de régularisation dimensionnelle. / In this thesis, we study the renormalization of Connes-Kreimer in the contex of specified Feynman graphs Hopf algebra. We construct a Hopf algebra structure $\mathcal{H}_\mathcal{T}$ on the space of specified Feynman graphs of a quantum field theory $\mathcal{T}$. We define also a doubling procedure for the bialgebra of specified Feynman graphs, a convolution product and a group of characters of this Hopf algebra with values in some suitable commutative algebra taking momenta into account. We then implement the renormalization described by A. Connes and D. Kreimer and the Birkhoff decomposition for two renormalization schemes: the minimal subtraction scheme and the Taylor expansion scheme.We recall the definition of Feynman integrals associated with a graph. We prove that these integrals are holomorphic in a complex variable D in the case oh Schwartz functions, and that they extend in a meromorphic functions in the case of a Feynman type functions. Finally, we determine the finite parts of Feynman integrals using the BPHZ algorithm after dimensional regularization procedure.

This thesis deals with out-of-equilibrium transport phenomena in strongly correlated low-dimensional fermionic systems, with special emphasis on π-conjugated molecular materials. The focus of this work is to study real-time dynamics of spin and charge transport in these systems in order to investigate non-equilibrium transport in single-molecule electronic and spintronic devices. Chapter 1 describes the electronic structure and dynamics of strongly correlated fermionic systems in general, and in one-dimension, in particular. For this purpose, effective low-energy model Hamiltonians (used in this work) are discussed. Whenever applicable, approximate analytical and numerical methods commonly used in the literature to deal with these model Hamiltonians, are outlined. In the context of one-dimensional strongly correlated fermionic systems, analytical techniques like the Bethe ansatz and bosonization, and numerical procedures like exact diagonalization and DMRG, used for solving finite systems, are discussed in detail. Chapter 2 provides an overview of the different zero-temperature (T = 0) time-dependent DMRG algorithms, which have been used to study out-of-equilibrium time-dependent phenomena in low-dimensional strongly correlated systems. In Chapter 3 we employ the time-dependent DMRG algorithm proposed by Luo, Xiang and Wang [Phys. Rev. Lett. 91, 049701 (2003)], to study the role of dimerization and electronic correlations on the dynamics of spin-charge separation. We employ the H¨uckel and Hubbard models for our studies. We have modified the algorithm proposed by Luo et. al to overcome some of its limitations. Chapter 4 presents a generalized adaptive time-dependent density matrix renormalization group (DMRG) scheme developed by us, called the Double Time Window Targeting (DTWT) technique, which is capable of giving accurate results with lesser computational resources than required by the existing methods. This procedure originates from the amalgamation of the features of pace keeping DMRG algorithm, first proposed by Luo et. al, [Phys.Rev. Lett. 91, 049701 (2003)], and the time-step targeting (TST) algorithm by Feiguin and White [Phys. Rev. B 72, 020404 (2005)]. In chapter 5 we apply the Double Time Window Targeting (DTWT) technique, which was discussed in the previous chapter, for studying real-time quantum dynamics of spin-charge separation in π-conjugated polymers. We employ the Pariser-Parr-Pople (PPP) model which has long-range electron-electron interactions. For investigating real-time dynamics of spin and charge transport, we inject a hole at one end of polyene chains of different lengths and study the temporal evolution of its spin and charge degrees of freedom, using the DTWT td-DMRG algorithm. Chapter 6 we investigate the effect of terminal substituents on the dynamics of spin and charge transport in donor-acceptor substituted polyenes (D- (CH)x- A) chains, also known as push-pull polyenes. We employ long-range correlated model Hamiltonian for the D- (CH)x- A system and, real-time DMRG dynamics for time propagating the wave packet obtained by injecting a hole at a terminal site in the ground state of the system. Our studies reveal that the end groups do not affect the spin and charge velocities in any significant way, but change the amount of charge transported. We have compared these with the polymethineimine (CN)x system in which besides electron affinities, the nature of pz orbitals in conjugation also alternate from site to site. Chapter 7 presents our investigation on the effect of static electron-phonon coupling (dimerization) on the dynamics of spin-charge separation in particular, and transport in general, in π-conjugated polyene chains. The polyenes are modeled by the Pariser-Parr-Pople Hamiltonian, having long-range electron-electron correlations. Our studies reveal that spin and charge velocities depend both on the chain length and dimerization. The spin and charge velocities increase as dimerization increases, but the amount of charge and spin transported along the chain decrease with enhancement in dimerization. Furthermore, in the range 0.3≤ δ≤0.5, it is observed that the dynamics of spin-charge separation becomes complicated, and the charge degree of freedom is affected more by electron-phonon coupling compared to the spin degree of freedom.

The concept of topology in condensed matter physics has led to the discovery of rich and exotic physics in recent years. Especially when strong correlations are included, phenomenons such as fractionalization and anyonic particle statistics can arise. In this thesis, we study several systems hosting topological phases of interacting fermions. In the first part, we consider one-dimensional systems of parafermions, which are generalizations of Majorana fermions, in the presence of a Z_N charge symmetry. We classify the symmetry-protected topological (SPT) phases that can occur in these systems using the projective representations of the symmetries and find a finite number of distinct phases depending on the prime factorization of N. The different phases exhibit characteristic degeneracies in their entanglement spectrum (ES). Apart from these SPT phases, we report the occurrence of parafermion condensate phases for certain values of N. When including an additional Z_N symmetry, we find a non-Abelian group structure under the addition of phases. In the second part of the thesis, we focus on two-dimensional lattice models of spinless fermions. First, we demonstrate the detection of a fractional Chern insulator (FCI) phase in the Haldane honeycomb model on an infinite cylinder by means of the density-matrix renormalization group (DMRG). We report the calculation of several quantities characterizing the topological order of the state, i.e., (i)~the Hall conductivity, (ii)~the spectral flow and level counting in the ES, (iii)~the topological entanglement entropy, and (iv)~the charge and topological spin of the quasiparticles. Since we have access to sufficiently large system sizes without band projection with DMRG, we are in addition able to investigate the transition from a metal to the FCI at small interactions which we find to be of first order. In a further study, we consider a time-reversal symmetric model on the honeycomb lattice where a Chern insulator (CI) induced by next-nearest neighbor interactions has been predicted by mean field theory. However, various subsequent studies challenged this picture and it was still unclear whether the CI would survive quantum fluctuations. We therefore map out the phase diagram of the model as a function of the interactions on an infinite cylinder with DMRG and find evidence for the absence of the CI phase. However, we report the detection of two novel charge-ordered phases and corroborate the existence of the remaining phases that had been predicted in mean field theory. Furthermore, we characterize the transitions between the various phases by studying the behavior of correlation length and entanglement entropy at the phase boundaries. Finally, we develop an improvement to the DMRG algorithm for fermionic lattice models on cylinders. By using a real space representation in the direction along the cylinder and a real space representation in the perpendicular direction, we are able to use the momentum around the cylinder as conserved quantity to reduce computational costs. We benchmark the method by studying the interacting Hofstadter model and report a considerable speedup in computation time and a severely reduced memory usage.

Experiments on neutrino oscillations have confirmed that neutrinos have small, but non-zero masses, and that the interacting neutrino states do not have definite masses, but are mixtures of such states.The seesaw models make up a group of popular models describing the small neutrino masses and the corresponding mixing.In these models, new, heavy fields are introduced and the neutrino masses are suppressed by the ratio between the electroweak scale and the large masses of the new fields. Usually, the new fields introduced have masses far above the electroweak scale, outside the reach of any foreseeable experiments, making these versions of seesaw models essentially untestable. However, there are also so-called low-scale seesaw models, where the new particles have masses above the electroweak scale, but within the reach of future experiments, such as the LHC.In quantum field theories, quantum corrections generally introduce an energy-scale dependence on all their parameters, described by the renormalization group equations. In this thesis, the energy-scale dependence of the neutrino parameters in two low-scale seesaw models, the low-scale type I and inverse seesaw models, are considered. Also, the question of whether the neutrinos are Majorana particles, \ie , their own antiparticles, has not been decided experimentally. Future experiments on neutrinoless double beta decay could confirm the Majorana nature of neutrinos. However, there could also be additional contributions to the decay, which are not directly related to neutrino masses. We have investigated the possible future bounds on the strength of such additional contributions to neutrinoless double beta decay, depending on the outcome of ongoing and planned experiments related to neutrino masses. / QC 20110812

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