Nesta dissertação estudaremos sistemas de bósons ultrafrios armadilhados em uma rede ótica quadrada bidimensional sem levar em consideração o confinamento harmônico. A dinâmica desses sistemas é bem descrita pelo modelo de Bose-Hubbard, que prevê uma transição de fase quântica de um superfluido para um isolante de Mott a temperaturas baixas, e pode ser induzida variando a profundidade do potencial da rede ótica. Apresentaremos o diagrama de fases dessa transição construído a partir de uma aproximação de campo médio e também com um cálculo numérico usando um algoritmo de Monte Carlo Quântico, denominado algoritmo Worm. Encontramos o ponto crítico para o primeiro lobo de Mott em ambos os casos, concordando com trabalhos anteriores. / This work study the two-dimensional ultracold bosonic atoms loaded in a square optical lattice, without harmonic confinement. The dynamics of this system is described by the Bose-Hubbard model, which predicts a quantum phase transition from a superfluid to a Mott-insulator at low temperatures that can be induced by varying the depth of the optical potential. We present here the phase diagram of this transition built from a mean field approach and from a numerical calculation using a Quantum Monte Carlo algorithm, namely the Worm algorithm. We found the critical transition point for the first Mott lobe in both cases, in agreement with the standard literature.

This thesis covers most of my work in the field of ultracold atoms loaded in optical lattices. This thesis can be divided into five different parts. In Chapter 1, after a brief introduction to the field of optical lattices I review the fundamental aspects pertaining to the physics of systems in periodic potentials and a short overview of the experiments on ultracold atoms in an optical lattice. In Chapter 2 we develop an inhomogeneous mean-field theory for the extended Bose-Hubbard model with a quadratic, confining potential. In the absence of this poten¬tial, our mean-field theory yields the phase diagram of the homogeneous extended Bose-Hubbard model. This phase diagram shows a superfluid (SF) phase and lobes of Mott-insulator(MI), density-wave(DW), and supersolid (SS) phases in the plane of the chemical potential and on-site repulsion ; we present phase diagrams for representative values of , the repulsive energy for bosons on nearest-neighbor sites. We demonstrate that, when the confining potential is present, superfluid and density-wave order parameters are nonuniform; in particular, we obtain, for a few representative values of parameters, spherical shells of SF, MI ,DW ,and SSphases. We explore the implications of our study for experiments on cold-atom dipolar con¬densates in optical lattices in a confining potential. In Chapter3 we present an extensive study of Mottinsulator( MI) and superfluid (SF) shells in Bose-Hubbard (BH) models for bosons in optical lattices with har¬monic traps. For this we develop an inhomogeneous mean-field theory. Our results for the BH model with one type of spinless bosons agrees quantitatively with quan¬tum Monte Carlo(QMC) simulations. Our approach is numerically less intensive than such simulations, so we are able to perform calculations on experimentally realistic, large three-dimensional(3D) systems, explore a wide range of parameter values, and make direct contact with a variety of experimental measurements. We also generalize our inhomogeneous mean-field theory to study BH models with har¬monic traps and(a) two species of bosons or(b) spin-1bosons. With two species of bosons we obtain rich phase diagrams with a variety of SF and MI phases and as¬sociated shells, when we include a quadratic confining potential. For the spin-1BH model we show, in a representative case, that the system can display alternating shells of polar SF and MI phases; and we make interesting predictions for experi¬ments in such systems. . In Chapter 4 we carry out an extensive study of the phase diagrams of the ex-tended Bose Hubbard model, with a mean filling of one boson per site, in one dimension by using the density matrix renormalization group and show that it contains Superfluid (SF), Mott-insulator (MI), density-wave (DW) and Haldane ¬insulator(HI) phases. We show that the critical exponents and central charge for the HI-DW,MI-HI and SF-MI transitions are consistent with those for models in the two-dimensional Ising, Gaussian, and Berezinskii-Kosterlitz-Thouless (BKT) uni¬versality classes, respectively; and we suggest that the SF-HI transition may be more exotic than a simple BKT transition. We show explicitly that different bound¬ary conditions lead to different phase diagrams.. In Chapter 5 we obtain the excitation spectra of the following three generalized of Bose-Hubbard(BH) models:(1) a two-species generalization of the spinless BH model, (2) a single-species, spin-1 BH model, and (3) the extended Bose-Hubbard model (EBH) for spinless interacting bosons of one species. In all the phases of these models we show how to obtain excitation spectra by using the random phase approximation (RPA). We compare the results of our work with earlier studies of related models and discuss implications for experiments.

While the free electron model can often be surprisingly successful in describing properties of solids, there are plenty of materials in which interactions between electrons are too significant to be neglected. These strongly correlated systems sometimes exhibit rather unexpected, unusual and useful phenomena, understanding of which is one of the aims of condensed matter physics. Heat capacity measurements of paramagnetic YFe$_{2}$Ge$_{2}$ give a Sommerfeld coefficient of about 100 mJ mol$^{−1}$ K$^{−2}$, which is about an order of magnitude higher than the value predicted by band structure calculations. This suggests the existence of strong electronic correlations in the compound, potentially due to proximity to an antiferromagnetic quantum critical point (QCP). Existence of the latter is also indicated by the non-Fermi liquid T$^{3/2}$ behaviour of the low temperature resistivity. Below 1.8 K a superconducting phase develops in the material, making it a rare case of a non-pnictide and non-chalcogenide iron based superconductor with the 1-2-2 structure. This thesis describes growth and study of a new generation of high quality YFe$_{2}$Ge$_{2}$ samples with residual resistance ratios reaching 200. Measurements of resistivity, heat capacity and magnetic susceptibility confirm the intrinsic and bulk character of the superconductivity, which is also argued to be of an unconventional nature. In order to test the hypothesis of the nearby QCP, resistance measurements under high pressure of up to 35 kbar have been conducted. Pressure dependence of the critical temperature of the superconductivity has been found to be rather weak. μSR measurements have been performed, but provided limited information due to sample inhomogeneity resulting in a broad distribution of the critical temperature. While the superconductivity is the result of an effective attraction between electrons, under different circumstances the electronic properties of a system can instead be dictated by the Coulomb repulsion. This is the case for another transition metal based compound NiS$_{2}$, which is a Mott insulator. Applying hydrostatic pressure of about 30 kbar brings the material across the Mott metal-insulator transition (MIT) into the metallic phase. We have used the tunnel diode oscillator (TDO) technique to measure quantum oscillations in the metallised state of NiS$_{2}$, making it possible to track the evolution of the principal Fermi surface and the associated effective mass as a function of pressure. New results are presented which access a wider pressure range than previous studies and provide strong evidence that the effective carrier mass diverges close to the Mott MIT, as expected within the Brinkman-Rice scenario and predicted in dynamical mean field theory calculations. Quantum oscillations have been measured at pressures as close to the insulating phase as 33 kbar and as high as 97 kbar. In addition to providing a valuable insight into the mechanism of the Mott MIT, this study has also demonstrated the potential of the TDO technique for studying materials at high pressures.

Submitted by Irene Nascimento (irene.kessia@ufpe.br) on 2017-04-17T18:48:07Z No. of bitstreams: 2 license_rdf: 1232 bytes, checksum: 66e71c371cc565284e70f40736c94386 (MD5) Dissertação Mestrado.pdf: 3421668 bytes, checksum: 4a6d6f2568841028e0b66b952f2d15ba (MD5) / Made available in DSpace on 2017-04-17T18:48:07Z (GMT). No. of bitstreams: 2 license_rdf: 1232 bytes, checksum: 66e71c371cc565284e70f40736c94386 (MD5) Dissertação Mestrado.pdf: 3421668 bytes, checksum: 4a6d6f2568841028e0b66b952f2d15ba (MD5) Previous issue date: 2016-08-30 / Este trabalho é uma extensão de uma modelagem tight−binding de estados de Wannier com interação coulombiana de screening para a descrição de elétrons correlacionados nas camadas de CuO2 do La2CuO4. Na condição de banda semicheia e temperatura T = 0 K, esta modelagem é capaz de predizer um estado fundamental antiferromagnético isolante e um estado excitado paramagnético com pseudo-gap, cuja natureza ainda é debatido na literatura. Esses estados são obtidos no contexto de uma solução auto-consistente tipo Hartree-Fock na modelagem de uma banda efetiva de estados de Wannier, sob condições de quebra de simetria de translações espaciais e efeito de emaranhamento (entanglement) na estrutura espinorial dos estados eletrônicos. No regime dopado com buracos, esses estados ficam degenerados num ponto crítico de concentração de buracos igual a xc = 0.2, resultando em uma transição de fase quântica de segunda ordem para um estado paramagnético. A modelagem dá assim explicação à existência detectada experimentalmente desta transição de fase. Em nosso trabalho generalizamos o termo cinético da modelagem acima mencionada através da inclusão de hopping entre segundos vizinhos, de acordo com observações experimentais. De fato, a inclusão deste novo termo cinético resulta em melhor concordância da previsão da modelagem e as observações experimentais da evolução da superfície de Fermi com dopagem de buracos. Em particular, enquanto na modelagem restrita a primeiros vizinhos não podemos conciliar a transição de fase quântica observada a xc = 0.2 com as características experimentais da evolução da superfície de Fermi, isto torna-se possível com a inclusão do hopping de segundos vizinhos com amplitude sugerida pelos resultados experimentais. / This work is an extension of a tight - binding model of states of Wannier with screened Coulomb interaction to the description of correlated electrons in the layers of CuO2 of the La2CuO4. In condition of half-filled band and temperature T = 0 K, this model is able to predict a antiferromagnetic insulating ground state and a paramagnetic excited state with pseudo-gap, which nature is still debated in the literature. These states are obtained in the context of a self-consistent solution type Hartree-Fock in the model of an effective band of the Wannier states under breaking conditions of symmetry of space translations and entanglement effect in the spinor structure of the electronic states. In scheme doped with holes, these states are degenerate at critical point of hole concentration equal to xc = 0.2, resulting in a quantum phase transition of second order to paramagnetic state. The model thus gives explanation to the existence of this experimentally detected phase transition. In our work we generalize the kinetic term of above mentioned model by inclusion of hopping between seconds neighboring, according to experimental observations. In fact, the inclusion of this new kinetic term results in better agreement of the forecast of the model and experimental observations of the evolution of the Fermi surface with holes doping. In particular, while the model is restricted to the first neighbors can not reconcile quantum phase transition observed at xc = 0.2 with the experimental characteristics of the evolution of the Fermi surface, this becomes possible with the inclusion of hopping between second neighboring with ampliude suggested by experimental results.

Through our whole work we study the XY model with all its entirety, a particular spin model where spins are confined in a plane. We try to bring out a good understanding of this model with all different types of phases and phase transition, it undergoes in critical situations. We conceive of these external conditions from very different physical models like High Tc Superconductor, Ultracold atoms in optical lattice which are in focus of recent research. Firstly we model high Tc Superconductors with very simple 2D XY model to get an idea about the diamagnetic response exhibited by these materials when kept in a external magnetic field. This modeling is reasonable because most of the physics of cuprate High Tc Superconductors are governed by their 2D copper oxide planes which insists us to consider 2D models. Later we shifted to a more realistic 3D anisotropic XY model , as the coupling between cuprates plane may have a considerable role in devising physics of those materials. We particularly focus on the 2D to 3D crossover effect on magnetisation showed by these models, with keeping an eye on how all these can be relate to the experimentally acquired magnetisation profile of High Tc Supercondutors. On the second project we investigate on the phase diagram of a fully frustrated 2-leg ladder Bose Hubbard model. After mapping it properly to a classical model, a bi-layer Fully Frustrated XY model on square lattice, we found that the frustration leads to the emergence of a new phase "Chiral Mott insulator(CMI)" sandwiched between "Chiral Superfluid(CSF)" and "regular Mott insulator(MI)" phase. We divide the whole report into four parts. The first chapter is basically contain-ing introductory part comprising the motivation. In the second chapter we discuss various types of phases and phase transitions of the 2D & 3D XY models. We try to address their critical behaviors. In the third chapter and onwards we consider our model in external magnetic field and observe magnetisation in these systems. Here we specially focus on 2D to 3D crossover effect on magtisation measurement. Lastly in the fourth chapter we bring out a correspondence of XY model with the 2 leg ladder fully frustrated Bose Hubbard Model. There we report the emergence of a new phase, Chiral Mott Insulator(CMI) due to frustration in system.

L'étude des phases de la matière est d'un intérêt fondamental en physique. La théorie de Landau, qui est le "modèle standard" des transitions de phases, caractérise les phases de la matière en termes des brisures de symétrie, décrites par un paramètre d'ordre local. Cette théorie a permis la description de phénomènes remarquables tels que la condensation de Bose-Einstein, la supraconductivité et la superfluidité.<p><p>Il existe cependant des phases qui échappent à la description de Landau. Il s'agit des phases quantiques topologiques. Celles-ci constituent un nouveau paradigme et sont caractérisées par un ordre global défini par un invariant topologique. Ce dernier classe les objets ou systèmes de la manière suivante: deux objets appartiennent à la même classe topologique s'il est possible de déformer continument le premier objet en le second. Cette propriété globale rend le système robuste contre des perturbations locales telles que le désordre. <p><p>Les atomes froids constituent une plateforme idéale pour simuler les phases quantiques topologiques. Depuis l'invention du laser, les progrès en physique atomique et moléculaire ont permis un contrôle de la dynamique et des états internes des atomes. La réalisation de gaz quantiques,tels que les condensats de Bose-Einstein et les gaz dégénérés de Fermi, ainsi que la réalisation de réseaux optiques à l'aide de faisceaux lasers, permettent d'étudier ces nouvelles phases de la matière et de simuler aussi la physique du solide cristallin.<p><p>Dans cette thèse, nous nous concentrons sur l'etude d'isolants topologiques avec des atomes froids. Ces derniers sont isolants de volume mais possèdent des états de surface qui sont conducteurs, protégés par un invariant topologique. Nous traitons trois sujets principaux. Le premier sujet concerne la génération dynamique d'un isolant topologique de Mott. Ici, les interactions engendrent l'isolant topologique et ce, sans champ de jauge de fond. Le second sujet concerne la détection des isolants topologiques dans les expériences d'atomes froids. Nous proposons deux méthodes complémentaires pour caractériser celles-ci. Finalement, le troisième sujet aborde des thèmes au-delà de la définition standard d'isolant topologique. Nous avons d'une part proposé un algorithme efficace pour calculer la conductivité de Berry, la contribution topologique à la conductivité transverse lorsque l'énergie de Fermi se trouve dans une bande d'énergie. D'autre part, nous avons utilisé des méthodes pour caractériser les propriétés quantiques topologiques de systèmes non-périodiques.<p><p>L'étude des isolants topologiques dans les expériences d'atomes froids est un sujet de recherche récent et en pleine expansion. Dans ce contexte, cette thèse apporte plusieurs contributions théoriques pour la simulation de systèmes quantiques sur réseau avec des atomes froids. / Doctorat en Sciences / info:eu-repo/semantics/nonPublished

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