Dissertations / Theses on the topic 'SUPERFLUID MODEL'

Recent experiments have been made with helium-3 immersed in a new type of aerogel. This aerogel consists of aluminium oxide strands, which are directed along roughly the same direction at macroscopic length scale. In this thesis we propose a simple theoretical model of this so called nematically ordered aerogel, and study the model using the Ginzburg-Landau theory of superfluidity. Because of the complexity of the theory, it is necessary to rely on numerical methods. We use two different iterative methods, Newton’s method and Conjugate Gradient method, to numerically minimise the Ginzburg-Landau free energy. Numerical simulations show that there are three different stable superfluid phases, A-like phase, B-like phase and polar phase. We determine the symmetries of these phases, because the symmetries provide a way to classify different phases. We then present phase diagrams for the model with different boundary conditions and parameters. We also try adding randomness to the model. A notable result is that the normal-superfluid transition is always from the normal phase to the polar phase. A comparison between the experimental phase diagram and the theoretical phase diagrams shows that the model can explain the experiments qualitatively, but not quantitatively.

In this thesis we present an extensive study on quantised vortex dynamics using the Gross-Pitaevskii model of a superfluid in the limit of zero temperature. We make use of an accurate and robust numerical method that we developed to detect topological defects present in the scalar order parameter characterising the superfluid. We begin by focusing on the scattering of vortex rings by a superfluid line vortex. Thereafter, we focus on the development and decay of a turbulent vortex tangle, measuring the Vinen’s decay law for the total vortex length. Moreover, the temporal evolution of the Kelvin wave spectrum is obtained providing evidence of the development of a weak-wave turbulence cascade. The study of superfluid vortex reconnections is also carried out in order to identify what aspects of the reconnection process are universal. Aside from the investigation on quantised vortex dynamics, in this thesis we also present a study on the motion of an electron bubble in a superfluid. The electron bubble dynamics is studied in the adiabatic approximation using the Gross-Pitaevskii equation to model the superfluid wavefunction and a Schro ̈dinger equation to model the electron wavefunction. This model allows us to recover the key dynamics of the ion-vortex interactions that arise and the subsequent ion-vortex complexes that can form. We determine the vortex-nucleation limited mobility of the ion to recover values in reasonable agreement with measured data. Moreover, considering the scenario of an ion trapped on the core of a vortex line, we investigate how small and large amplitude Kelvin waves and solitary waves affect the drift velocity of the ion. In particular, we have identified that Hasimoto soliton-bubble complexes propagating along the vortex can arise.

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.

Les systèmes avioniques s'étoffent et se complexifient de plus en plus. Avec l'augmentation des capacités de calcul, de nouvelles architectures basées sur le partage de ressources émergent. Effectuer le diagnostic d'un système n'est désormais plus une opération anodine. L'enjeu actuel est donc de mettre en place des techniques de diagnostic performantes tout en optimisant les capacités de monitoring nécessaires.Ce mémoire donne une caractérisation basée modèle d'un système sous diagnostic, puis propose des techniques pour en évaluer les performances de diagnostic, ainsi que celles de son monitoring (relativement à ces performances). Le contexte industriel dans lequel s'inscrit cette thèse amène d'autres contraintes, notamment la prise en compte de la taille des systèmes avioniques à analyser. Cette thèse étudie alors l'applicabilité des techniques introduites dans ce contexte et en propose une adaptation
Avionics systems become more and more complex. With the improvment of computing possibilities, new architectures based on resources sharing are growing up. Perform diagnosis of a system is no longer a trivial operation. The challenge is to develop efficient techniques of diagnosis while optimizing capabilities of monitoring required.This thesis give a model-based characterization of a system under diagnosis, and proposes techniques to assess diagnostic performances, as well as its monitoring ones (with respect to these diagnostic performances). The industrial context of this thesis brings other constraints, and in particular the need to handle the size of avionics systems to analyze. That thesis then examines the applicability of the introduced techniques to this particular context, and proposes an adaptation

Cette thèse s'intéresse aux phénomènes électrostatiques émergents dans les modèles magnétiques toroïdaux bi-dimensionnels à symétrie XY, fournissant ainsi un support pour de plus amples recherches dans le domaine de la transition de phase Berezinskii-Kosterlitz-Thouless (BKT).Dans de nombreux systèmes bi-dimensionnels, dont le modèle bi-dimensionnel XY du magnétisme, la transition BKT contrôle la dissociation thermique de paires de défauts topologiques liés. Le modèle XY est analogue au gaz de Coulomb bi-dimensionnel, à ceci près qu'il peut être simulé sans avoir à modéliser les interactions à longue distance du système Coulombien. Cette thèse élucide ce paradoxe en démontrant que l'approximation de Villain appliquée au modèle XY est strictement équivalente au modèle électrostatique de Maggs-Rossetto (MR) appliqué au système Coulombien bi-dimensionnel.Cette équivalence est utilisée pour sonder la transition BKT par l'application de l'algorithme MR au gaz de Coulomb bi-dimensionel. En simulant le système Coulombien, il est prouvé que les fluctuations dans l'organisation des charges autour du tore sont activées à la température de transition BKT. Ces fluctuations du champ électrique indiquent ainsi la phase de haute température de la transition.Il est ensuite montré que l'exposant critique effectif de la théorie de Bramwell-Holdsworth (BH) peut être mesuré dans les films d'hélium 4 superfluide, qui correspondent à des gaz de Coulomb effectifs dans la limite de systèmes de grandes tailles finies
This thesis addresses the emergent electrostatics of two-dimensional, toroidal magnetic models that possess XY symmetry, providing a platform for novel investigations into the Berezinskii-Kosterlitz-Thouless (BKT) phase transition.The BKT transition drives the thermal dissociation of bound pairs of topological defects in many two-dimensional systems, including the two-dimensional XY model of magnetism. The XY model is closely analogous to the two-dimensional Coulomb gas, but can be simulated without computing the long-range interactions of the Coulombic system. This thesis elucidates this paradox by showing that Villain's approximation to the XY model is strictly equivalent to the Maggs-Rossetto (MR) electrostatic model when applied to the two-dimensional Coulomb gas.The mapping is used to probe the BKT transition through the application of the MR algorithm to the two-dimensional Coulomb gas. By simulating the Coulombic system, fluctuations in the winding of charges around the torus are shown to turn on at the BKT transition temperature. These topological-sector fluctuations in the electric field therefore signal the high-temperature phase of the transition.It is then shown that the effective critical exponent of Bramwell-Holdsworth (BH) theory can be measured in superfluid 4He films, which correspond to effective Coulomb gases in the limit of large but finite system size. With the Coulombic system taken as the base BKT system, it is inferred that BH theory is a general property of BKT systems

L'objectif de cette thèse est d'étudier différents aspects microscopiques et macroscopiques liés à la présence de superfluidité dans les étoiles à neutrons. Dans un premier temps, nous avons calculé des configurations stationnaires d'étoiles à neutrons superfluides en rotation, en relativité générale, basées sur l'utilisation d'équations d'état réalistes. A l'aide de ces configurations d'équilibre, nous avons ensuite développé un modèle simple de glitch, en relativité générale, vu comme un transfert de moment cinétique entre les neutrons superfluides et les particules chargées constituant l'étoile. Cela nous a permis d'obtenir des temps caractéristiques de montée qui pourront être comparés à de futures observations précises de glitches afin d'apporter de meilleures contraintes sur l'intérieur de ces étoiles. Enfin, nous nous sommes également intéressés à la dynamique des vortex superfluides, en présence de tubes de flux, dans le cas où les protons dans le coeur des étoiles formeraient un supraconducteur de type II
The aim of this thesis is to study different aspects, both microscopic and macroscopic, associated with the presence of a large amount of superfluid matter inside neutron stars. First, we computed stationary configurations of rotating superfluid neutron stars, in general relativity, using realistic equations of state. Based on these equilibrium configurations, we then developed a simple model of pulsar glitches, in general relativity, seen as angular momentum transfers between the superfluid neutrons and the charged particles composing the star. This enables us to infer spin-up time scales that could be compared with future accurate glitch observations, in order to get some constraints on the interior of neutron stars. Finally, we also focused on the dynamics of superfluid vortex lines, accounting for the presence of fluxtubes, if the protons are forming a type II superconductor in the core of neutron stars

Cette thèse présente la production d'un gaz dégénéré de rubidium 87 dans le régime quasi bidimensionnel (2D) et l'étude des modes collectifs de ce gaz. Nous montrons que le gaz quasi-2D peut être amené en dessous du seuil de la transition Berezinskii-Kosterlitz-Thouless. Nous montrons le caractère superfluide du gaz dégénéré par la présence des modes quadrupolaire et ciseaux, dont nous mesurons les fréquences d'oscillation. Son caractère bidimensionnel est vérifié par la mesure de la fréquence du mode monopolaire. Nous mettons en évidence l'influence du confinement transverse et de la troisième dimension sur la fréquence de ce mode. Pour produire le superfluide, un condensat de Bose-Einstein est d'abord produit dans un piège quadrupolaire bouché par un faisceau laser très désaccordé et soigneusement optimisé pour réduire les pertes Majorana par renversement de spin. Le condensat est ensuite transféré vers un "piège habillé", c'est-à-dire un potentiel adiabatique dans lequel les atomes sont habillés par un champ radiofréquence. Pour rendre le piège plus anisotrope, le gradient magnétique est augmenté au maximum, ce qui nous permet d'explorer le régime quasi-2D pour le gaz de Bose. Les deux types de piège utilisés sont caractérisés en détail. Nous tirons parti de la souplesse du potentiel adiabatique pour exciter et étudier les modes collectifs.

Condensed matter systems undergoing phase transitions rarely allow exact solutions. The presence of disorder renders the situation  even worse but collective Monte Carlo methods and parallel algorithms allow numerical descriptions. This thesis considers classical phase transitions in disordered spin systems in general and in effective models of superfluids with disorder and novel interactions in particular. Quantum phase transitions are considered via a quantum to classical mapping. Central questions are if the presence of defects changes universal properties and what qualitative implications follow for experiments. Common to the cases considered is that the disorder maps out correlated structures. All results are obtained using large-scale Monte Carlo simulations of effective models capturing the relevant degrees of freedom at the transition. Considering a model system for superflow aided by a defect network, we find that the onset properties are significantly altered compared to the $\lambda$-transition in $^{4}$He. This has qualitative implications on expected experimental signatures in a defect supersolid scenario. For the Bose glass to superfluid quantum phase transition in 2D we determine the quantum correlation time by an anisotropic finite size scaling approach. Without a priori assumptions on critical parameters, we find the critical exponent $z=1.8 \pm 0.05$ contradicting the long standing result $z=d$. Using a 3D effective model for multi-band type-1.5 superconductors we find that these systems possibly feature a strong first order vortex-driven phase transition. Despite its short-range nature details of the interaction are shown to play an important role. Phase transitions in disordered spin models exposed to correlated defect structures obtained via rapid quenches of critical loop and spin models are investigated. On long length scales the correlations are shown to decay algebraically. The decay exponents are expressed through known critical exponents of the disorder generating models. For cases where the disorder correlations imply the existence of a new long-range-disorder fixed point we determine the critical exponents of the disordered systems via finite size scaling methods of Monte Carlo data and find good agreement with theoretical expectations.

QC 20150306

We study the phase transition super uid-Mott insulator in optical lattices through model Jaynes-Cummings-Hubbard using the fermion approximation. Initially we did a review of the Jaynes-Cummings model that is a model that describes the interaction between an atom with two levels and a quantized electromagnetic eld. Then we stu- died the properties of the phase transition super uid-Mott insulator (SF-MI) through the model Bose-Hubbard and we discuss the characteristics of the two quantum phases and the conditions for phase transition. Apply the model Jaynes-Cummings-Hubbard in order to study the in uence of topology of network Bravais linear, square, single cubic (SC), body-centered cubic (BCC) and face-centered cubic (FCC) in the phase transi- tion SF-MI to different numbers of excitation and different detuning values between the atom and the eld and we found that the Mott lobes and the critical hopping terms are not renormalizable only for the FCC network. But the regime from many excitations, the critical hopping is renormalizable to all networks and is independent of detuning. Then we study the Jaynes-Cummings-Hubbard model for a chain with the addition of the Kerr effect (nonlinear optical effect) through the approximation of fermions (FA). We observed that the Kerr effect does not cause large changes in the energy spectrum. However, the properties of the phase transition SF-MI undergoes signicant changes due to the Kerr effect. Another important feature which has been observed is that the Kerr effect favors the phase MI.
Estudamos a transi¸c ao de fase superfluido-isolante de Mott (SF-MI) em redes ´opticas via modelo Jaynes-Cummings-Hubbard utilizando a aproxima¸c ao de f´ermions. Inicialmente fazemos uma revis ao do modelo Jaynes-Cummings que descreve a intera¸c ao de um ´atomo de dois n´ıveis com um campo eletromagn´etico quantizado. Em seguida estudamos as propriedades da transi¸c ao de fase SF-MI no modelo Bose-Hubbard e discutimos as caracter´ısticas das duas fase qu anticas e as condi¸c oes para a transi¸c ao de fase. Aplicamos o modelo Jaynes-Cummings-Hubbard com a finalidade de estudarmos a influ encia da topologia das redes de Bravais linear, quadrada, c´ubica simples (SC), c´ubica de corpo centrado (BCC) e c´ubica de faces centradas (FCC) na transi¸c ao de fase SF-MI para diferentes n´umeros de excita¸c ao e diferentes valores de dessintonia entre o ´atomo e o campo e encontramos que os l´obulos de Mott n ao s ao renormaliz ´aveis apenas para a rede FCC. Mas no regime de muitas excita¸c oes, as redes BCC e FCC passam a ter o mesmo comportamento enquanto que as demais continuam com sua depend encia no n´umero de vizinhos. Em seguida, estudamos o modelo de Jaynes-Cummings-Hubbard para uma cadeia com a adi¸c ao do efeito tipo Kerr (efeito ´optico n ao-linear) atrav´es da aproxima¸c ao de f´ermions (FA). Observamos que o efeito tipo Kerr n ao provoca grandes altera¸c oes no espectro de energia. No entanto, as propriedades de transi¸c ao de fase SF-MI passam por mudan¸cas significativas devido ao efeito tipo Kerr. Outra caracter´ıstica importante que foi observada ´e que o efeito tipo Kerr favorece a fase MI.

Investigamos a transição de localização induzida por desordem em condensados de Bose- Einstein em redes unidimensionais, no limite não interagente, utilizando os modelos de Anderson e de Aubry-Andr´e para a desordem. Através de diagonalização numérica exata, mostramos que, além da fração de superfluído, usualmente empregada, outras ferramentas, como o emaranhamento e a fidelidade, são indicadores claros da transição. Os valores críticos que encontramos para a amplitude da desordem estão de acordo com os resultados conhecidos para a transição de Anderson nos dois modelos. É interessante ressaltar que a fidelidade exibe boa precisão mesmo para redes pequenas. Efeitos do tamanho do sistema são analisados em detalhe para os dois modelos. Isto inclui a determinação de uma lei de escala de tamanho finito para a amplitude de desordem crítica no modelo de Anderson. Já no modelo de Aubry-André observamos que efeitos de tamanho finito são muito menos pronunciados, estando basicamente associados a diferenças entre tamanho de rede e periodicidade do potencial.
We investigate the disorder-induced localization transition in Bose-Einstein condensates on one-dimensional lattices, in the non-interacting limit, utilizing the Anderson and Aubry- Andr´e models for disorder. Using exact numerical diagonalization, we show that, in addition to the standard superfluid fraction, other tools such as the entanglement and fidelity can provide clear signatures of the transition. The critical values we find for the disorder amplitude are in agreement with known results for the Anderson transition in both models. Interestingly, the fidelity exhibits good sensitivity even for small lattices. System-size effects are analyzed in detail for both models. This includes the determination of a finite-size-scaling law for the critical disorder strength in the case of the Anderson model. For the Aubry- Andr´e model we observe that finite-size effects are much less pronounced, being basically associated with the mismatching between lattice size and periodicity of the potential.

In this thesis we study a variety of problems in superfluid turbulence, princi-pally in two dimensions. A summary of the main results of our studies is given below; we indicate the Chapters in which we present these. In Chapter 1, we provide an overview of several problems in superfluid turbulence with special emphasis on background material for the problems we study in this thesis. In particular, we give: (a) a brief introduction of fluid turbulence; (b) an overview of superfluidity and the phenomenological two-fluid model; (c) a brief overview of experiments on superfluid turbulence; (d) an introductory accounts of the phenomenological models used in the study of superfluid turbulence. We end with a summary of the problems we study in subsequent Chapters of this thesis. In Chapter 2, we present a systematic, direct numerical simulation of the two-dimensional, Fourier-truncated, Gross-Pitaevskii equation to study the turbulent evolutions of its solutions for a variety of initial conditions and a wide range of parameters. We find that the time evolution of this system can be classified into four regimes with qualitatively different statistical properties. First, there are transients that depend on the initial conditions. In the second regime, power- law scaling regions, in the energy and the occupation-number spectra, appear and start to develop; the exponents of these power laws and the extents of the scaling regions change with time and depend on the initial condition. In the third regime, the spectra drop rapidly for modes with wave numbers k > kc and partial thermalization takes place for modes with k < kc ; the self-truncation wave number kc(t) depends on the initial conditions and it grows either as a power of t or as log t. Finally, in the fourth regime, complete thermalization is achieved and, if we account for finite-size effects carefully, correlation functions and spectra are consistent with their nontrivial Berezinskii-Kosterlitz-Thouless forms. Our work is a natural generalization of recent studies of thermalization in the Euler and other hydrodynamical equations; it combines ideas from fluid dynamics and turbulence, on the one hand, and equilibrium and nonequilibrium statistical mechanics on the other. In Chapter 3, we present the first calculation of the mutual-friction coefficients α and α (which are parameters in the Hall-Vinen-Bekharevich-Khalatnikov two-fluid model that we study in chapter 5) as a function of temperature in a homogeneous Bose gas in two-dimensions by using the Galerkin-truncated Gross-Pitaevskii equation, with very special initial conditions, which we obtain by using the advective, real, Ginzburg-Landau equation (ARGLE) and an equilibration procedure that uses a stochastic Ginzburg-Landau equation (SGLE). We also calculate the normal-fluid density as a function of temperature. In Chapter 4, we elucidate the interplay of particles and fields in superfluids, in both simple and turbulent flows. We carry out extensive direct numerical simulations (DNSs) of this interplay for the two-dimensional (2D) Gross-Pitaevskii (GP) equation. We obtain the following results: (1) the motion of a particle can be chaotic even if the superfluid shows no sign of turbulence; (2) vortex motion depends sensitively on particle charateristics; (3) there is an effective, superfluid-mediated, attractive interaction between particles; (4) we introduce a short-range repulsion between particles, with range rSR, and study two- and many-particle collisions; in the case of two-particle, head-on collisions, we find that, at low values of rSR, the particle collisions are inelastic with coefficient of restitution e = 0; and, as we in-crease rSR, e becomes nonzero at a critical point, and finally attains values close to 1; (5) assemblies of particles and vortices show rich, turbulent, spatio-temporal evolution. In Chapter 5, we present results from our direct numerical simulations (DNSs) of the Hall-Vinen-Bekharevich-Khalatnikov (HVBK) two-fluid model in two dimensions. We have designed these DNSs to study the statistical properties of inverse and forward cascades in the HVBK model. We obtain several interesting results that have not been anticipated hitherto: (1) Both normal-fluid and superfluid energy spectra, En(k) and Es(k), respectively, show inverse- and forward-cascade regimes; the former is characterized by a power law Es(k) En(k) kα whose exponent is consistent with α 5/3. (2) The forward-cascade power law depends on (a) the friction coefficient, as in 2D fluid turbulence, and, in addition, on (b) the coefficient B of mutual friction, which couples normal and superfluid compo-nents. (3) As B increases, the normal and superfluid velocities, un and us, re-spectively, get locked to each other, and, therefore, Es(k) En(k), especially in the inverse-cascade regime. (4) We quantify this locking tendency by calculating the probability distribution functions (PDFs) P(cos(θ)) and P(γ), where the angle θ ≡ (un • us)/( |un||us|) and the amplitude ratio γ = |un|/|us |; the former has a peak at cos(θ) = 1; and the latter exhibits a peak at γ = 1 and power-law tails on both sides of this peak. (4) This locking increases as we increase B, but the power-law exponents for the tails of P(γ) are universal, in so far as they do not depend on B, ρn/ρ, and the details of the energy-injection method. (5) We characterize the energy and enstrophy cascades by computing the energy and enstrophy fluxes and the mutual-friction transfer functions for all wave-number scales k. In Chapter 6, we examine the multiscaling of structure functions in three-dimensional superfluid turbulence by using a shell-model for the three-dimensional HVBK equations. Our HVBK shell model is based on the GOY shell model. In particular, we examine the dependence of multiscaling on the normal-fluid fraction and the mutual-friction coefficients. We hope our in silico studies of 2D and 3D superfluid turbulence will stimulate new experimental, numerical, and theoretical studies.

This Thesis is devoted to the investigation of superfluid helium flows due to torsional oscillators. In its first part, flow due to a torsionally oscillating disc suspended on a tungsten filament is studied (building upon the work of A. C. Hollis Hallett from 1952). Measurements of the motion of the torsionally oscillating disc were performed in superfluid helium at temperatures between 1.265 K and 2.157 K at saturated vapour pressure. Time traces of the disc angular deflection were obtained, and critical parameters related to the turbulent flow stability were determined. In laminar flow, scaling of drag forces with the dimensionless Donnelly number was verified. Based on these results, and comparison with the original work, a scenario of the decay of turbulent flow was suggested. The second part of this work is focused on the development and construction of a similar experiment for mK temperatures. A new type of oscillator was designed, a so-called "pillbox", and a series of testing measurements was performed both at room and mK temperatures.

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.

The advancement in the eld of cold-atoms has generated a lot of interest in the condensed matter community. Cold-atom experiments can simulate clean, disor-der/impurity free systems very easily. In these systems, we have a control over various parameters like tuning the interaction between particles by the Feshbach resonance, tuning the hopping between lattice sites by laser intensity and so on. As a result, these systems can be used to mimic various theoretical models, which was hindered because of various experimental limitations. Thus, we have an ex-perimental tool in which we can start with a simple theoretical model and later tune the model experimentally and theoretically to simulate the real materials. This will be helpful in studying the physics of the real materials as we can control interactions as well as the impurities can also be taken care of. But the advance-ment in the eld of cold atoms has seen a roadblock for the fermions in optical lattices. The super uid and anti-ferromagnetic phases has not been achieved for fermions in optical lattices due to the \cooling problem" (entropy issues). In this thesis, we have addressed the issue of the \cooling problem" for fermions in optical lattice systems and studied the system with determinant quantum Monte Carlo technique. We start by giving a general idea of cold-atoms and optical lat-tice potentials, and a brief review of the experimental work going on in the cold-atomic systems. Experimental limitations like \fermionic cooling problem" have been discussed in some detail. Then we proposed a bilayer band-insulator model to circumvent the \entropy problem" and simultaneously increasing the transi-tion temperature for fermions in optical lattices. We have studied the attractive Hubbard model, which is the minimal model for fermions in optical lattices. The techniques that we have used to study the model are mean- eld theory, Gaussian uctuation theory and determinant quantum Monte Carlo numerical technique. . Chapter-1 : provides a general introduction to the ultra-cold atoms, optical lattice and Feshbach resonance. In this chapter we have discussed about cold-atom experiments in optical lattice systems. Here, we have brie y discussed the control over various parameters in the experiments. The goal of these experiments is to realize or mimic many many-body Hamiltonians in experiments, which until now was just a theoretical tool to describe various many-body physics. In the end we give a brief idea for introducing disorder in the cold-atom experiments discuss the limitations of these experiments in realizing the \interesting" super uid and anti-ferromagnetic phases of fermionic Hubbard model in optical lattices. Chapter-2 : gives a brief idea of \Determinant Quantum Monte-Carlo" (DQM C) technique that has been used to study these systems. In this chapter we will discuss the DQM C algorithm and the observables that can be calculated. We will discuss certain limitation of the DQM C algorithm like numerical instability and sign problem. We will brie y discuss how sign problem doesn't occur in the model we studied. Chapter-3 : discusses the way by which we can bypass the \cooling problem" (high entropy state) to get a fermionic super uid state in the cold atom experi-ments. In this chapter we propose a model whose idea hinges on a low-entropy band-insulator state, which can be tuned to super uid state by tuning the on-site attractive interaction by Feshbach resonance. We show through Gaussian uctua-tion theory that the critical temperature achieved is much higher in our model as compared to the single-band Hubbard model. Through detailed variational Monte Carlo calculations, we have shown that the super uid state is indeed the most stable ground state and there is no other competing order. In the end we give a proposal for its realization in the ultra-cold atom optical lattice systems. Chapter-4 : discusses the DQM C study of the model proposed in chapter- 3. Here we have studied the various single-particle properties like momentum distribution, double occupancies which can be easily measured in cold-atom ex-periments. We also studied the pair-pair and the density-density correlations in detail through DQM C algorithm and mapped out the full T U phase diagram. We show that the proposed model doesn't favor the charge density wave for the interaction strengths we are interested in. Chapter-5 : gives a brief idea of the e ect of adding an on-site random disorder in our proposed bilayer-Hubbard model. We study the e ect of random disorder on various single-particle properties which can be easily veri ed in cold-atom ex-periments. We studied the suppression of the pair-pair correlations as we increase the disorder strength in our proposed model. We nd that the critical value of the interaction doesn't change in the weak-disorder limit. We estimated the critical disorder strength needed to destroy the super uid state and argued that the tran-sition from the super uid to Bose-glass phase in presence of disorder lies in the universality class of (d + 1) XY model. In the end, we give a schematic U V phase diagram for our system. Chapter-6 : We studied the bilayer attractive Hubbard model in different lattice geometry, the bilayer honeycomb lattice, both in presence and in absence of the on-site random disorder. We discussed how the pair-pair and density-density cor-relations behave in the presence and absence of disorder. Through the finite-size scaling analysis we see the co-existence of the super fluid and the charge density wave order at half- lling. An in nitesimal disorder destroys the CDW order com-pletely while the super uid phase found to be robust against weak-disorder. We estimated the critical interaction strength, the critical temperature and the critical disorder strength through nite-size scaling, and provide a putative phase diagram for the system considered.

by Lin Lap-Ming.
Thesis (M.Phil.)--Chinese University of Hong Kong, 1999.
Includes bibliographical references (leaves [105]-109).
Text in English; abstracts in English and Chinese.
by Lin Lap-Ming.
Abstract --- p.i
Acknowledgement --- p.ii
Contents --- p.iii
List of Figures --- p.vi
List of Tables --- p.ix
Chapter Chapter 1. --- Introduction --- p.1
Chapter 1.1 --- Physical Motivation --- p.1
Chapter 1.2 --- Quasi-Normal Modes --- p.3
Chapter 1.3 --- Superfluidity in Neutron Stars --- p.7
Chapter 1.4 --- Outline of this Thesis --- p.9
Chapter Chapter 2. --- The Ordinary Perfect Fluid Neutron Stars --- p.11
Chapter 2.1 --- The Equilibrium Neutron Star Models --- p.11
Chapter 2.2 --- Non-Radial Oscillations of Neutron Stars --- p.14
Chapter 2.3 --- The Quasi-Normal Modes of Stars --- p.17
Chapter 2.3.1 --- The Fluid Modes --- p.17
Chapter 2.3.2 --- The Spacetime Modes --- p.18
Chapter Chapter 3. --- The General Relativistic Superfluid Formalism --- p.22
Chapter 3.1 --- The Carter Formalism --- p.22
Chapter 3.2 --- The Master Function --- p.25
Chapter Chapter 4. --- The Equilibrium Superfluid Neutron Stars --- p.27
Chapter 4.1 --- The Equilibrium Configurations --- p.27
Chapter 4.2 --- Initial and Boundary Conditions --- p.34
Chapter 4.3 --- Polytropic Models --- p.36
Chapter Chapter 5. --- Non-Radial Oscillations of Superfluid Neutron Stars --- p.40
Chapter 5.1 --- The Linearized Field Equations Inside the Star --- p.40
Chapter 5.1.1 --- Equations for Even-Parity Perturbations --- p.45
Chapter 5.1.2 --- Equations for Odd-Parity Perturbations --- p.48
Chapter 5.2 --- Initial and Boundary Conditions --- p.49
Chapter 5.2.1 --- Radial Integration Initial Conditions --- p.49
Chapter 5.2.2 --- Boundary conditions at the Surface --- p.55
Chapter 5.3 --- The Linearized Field Equations Outside the Star --- p.57
Chapter 5.4 --- Numerical Technique --- p.60
Chapter Chapter 6. --- Quasi-Normal Modes Extraction --- p.62
Chapter 6.1 --- Numerical Techniques for Quasi-Normal Modes Extraction --- p.62
Chapter 6.2 --- The Leaver Series --- p.64
Chapter 6.3 --- The Graphical Method --- p.67
Chapter Chapter 7. --- The Quasi-Normal Modes of Superfluid Neutron Stars --- p.69
Chapter 7.1 --- Polytropic Models --- p.69
Chapter 7.1.1 --- The w-modes --- p.70
Chapter 7.1.2 --- The f- and p-modes --- p.74
Chapter 7.2 --- Ideal Neutron-Proton-Electron Gas --- p.82
Chapter 7.3 --- Convergence Tests and Accuracy --- p.92
Chapter Chapter 8. --- Conclusion --- p.95
Appendix A. Speeds of Sound --- p.97
Appendix B. Equations for Radial Oscillations --- p.99
Appendix C. Numerical Technique for Solving Leaver's Series --- p.101
Appendix D. Scaling in Numerical Calculations --- p.103
Bibliography --- p.105