In this dissertation, we demonstrate the importance of invariant objects in many areas of applied research. The areas of application we consider are chemistry, celestial mechanics and aerospace engineering, plasma physics, and coupled map lattices. In the context of chemical reactions, stable and unstable manifolds of fixed points separate regions of phase space that lead to a certain outcome of the reaction. We study how these regions change under the influence of exposing the molecules to a laser. In celestial mechanics and aerospace engineering, we compute periodic orbits and their stable and unstable manifolds for a object of negligible mass (e.g. a satellite or spacecraft) under the presence of Jupiter and two of its moons, Europa and Ganymede. The periodic orbits serve as convenient spot to place a satellite for observation purposes, and computing their stable and unstable manifolds have been used in constructing low-energy transfers between the two moons. In plasma physics, an important and practical problem is to study barriers for heat transport in magnetically confined plasma undergoing fusion. We compute barriers for which heat cannot pass through. However, such barriers break down and lead to robust partial barriers. In this latter case, heat can flow across the barrier, but at a very slow rate. Finally, infinite dimensional coupled map lattice systems are considered in a wide variety of areas, most notably in statistical mechanics, neuroscience, and in the discretization of PDEs. We assume that the interaction amont the lattice sites decays with the distance of the sites, and assume the existence of an invariant whiskered torus that is localized near a collection of lattice sites. We prove that the torus has invariant stable and unstable manifolds that are also localized near the torus. This is an important step in understanding the global dynamics of such systems and opens the door to new possible results, most notably studying the problem of energy transfer between the sites. / text

We investigate the quantum dynamics of the spins on different Heisenberg antiferromagnetic spin lattice systems. Firstly, we applied the coupled-cluster method to the spin-1/2 antiferromagnetic XXZ model on a square lattice by employing an approximation which contains two-body long-range correlations and high-order four-body local correlations. Improvement is found for the ground-state energy, sublattice magnetization, and the critical anisotropy when comparing with the approximation including the two-body correlations alone. We also obtain the full excitation spectrum which is in good agreement with the quantum Monte Carlo results and the high-order spin-wave theory. Secondly, we study the longitudinal excitations of quantum antiferromagnets on a triangular lattice by a recently proposed microscopic many-body approach based on magnon-density waves. We calculate the full longitudinal excitation spectra of the antiferromagnetic Heisenberg model for a general spin quantum number in the isotropic limit. Similar to the square lattice model, we find that, at the center of the first hexagonal Brillouin zone Γ(q=0) and at the magnetic ordering wavevectors ±[Q= (4π/3,0)], the excitation spectra become gapless in the thermodynamic limit, due to the slow, logarithmic divergence of the structure factor. However, these longitudinal modes on two-dimensional models may be considered as quasi-gapped, as any finite-size effect or small anisotropy will induce a large energy gap, when compared with the counterpart of the transverse spin-wave excitations. We have also investigated the excited states of the quasi-one-dimensional quantum antiferromagnets on hexagonal lattices, including the longitudinal modes based on the magnon-density waves. A model Hamiltonian with a uniaxial single-ion anisotropy is first studied by a spin-wave theory based on the one-boson method; the ground state thus obtained is employed for the study of the longitudinal modes. The full energy spectra of both the transverse modes (i.e., magnons) and the longitudinal modes are obtained as functions of the nearest-neighbor coupling and the anisotropy constants. We have found two longitudinal modes due to the non-collinear nature of the triangular antiferromagnetic order, similar to that of the phenomenological field theory approach by Affleck. The excitation energy gaps due to the anisotropy and the energy gaps of the longitudinal modes without anisotropy are then investigated. We then compares our results for the longitudinal energy gaps at the magnetic wavevectors with the experimental results for several antiferromagnetic compounds with both integer and non-integer spin quantum numbers, and we find good agreements after the higher-order contributions are included in our calculations.

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.

Cold bosonic atoms trapped in optical lattices formed by standing wave interference patterns of multiple laser beams constitute excellent emulators of models of strongly correlated quantum systems of bosons. In this thesis, we develop and deploy strong-coupling expansion (i.e., an expansion in terms of the ratio of the inter-site hopping amplitude of the bosons to the strength of their interactions) techniques for studying the properties of three different instances of such systems. In the first instance, we have used strong coupling expansion techniques to calculate the density pro le for bosonic atoms trapped in an optical lattice with an overall harmonic trap at finite temperatures and large on site interaction in the presence of super fluid regions. Our results match well with quantum Monte Carlo simulations at finite temperature. We present calculations for the entropy per particle as a function of temperature which can be used to calibrate the temperature in experiments. Our calculations for the scaled density in the vacuum-to-super fluid transition agree well with the experimental data for appropriate temperatures. We also discuss issues connected with the demonstration of universal quantum critical scaling in the experiments. Experimental realizations of “atomtronic" Josephson junctions have recently been created in annular traps in relative rotation with respect to potential barriers that generate the weak links. If these devices are additionally subjected to optical lattice potentials, then they can incorporate strong-coupling Mott physics within the design, which can modify the behaviour and can allow for interesting new configurations of system generated barriers and of super fluid ow patterns. we have examined theoretically the behavior of a Bose super fluid in an optical lattice in the presence of an annular trap and a barrier across the annular region which acts as a Josephson junction. As the fluid is rotated relative to the barrier, it generates circulating super-currents until, at larger speeds of rotation, it develops phase slips which are typically accompanied by vortices. We use a finite temperature strong-coupling expansion about the mean- held solution of the Bose Hubbard model to calculate various properties of the device. In addition, we discuss some of the rich behavior that can result when there are Mott regions within the system. Rubidium-Cesium dipolar molecule formation through Feshbach resonance is an area of great current interest, for, the dipolar molecules, once formed, interact via v long range dipolar forces, leading to possibilities of novel phases. Experimentalists currently make such systems mostly using trial and error, and the resulting efficiencies for molecule formation tend to be low. With a goal to assist cold-atom experimentalists to achieve higher e ciencies of molecule formation, we have estimated the trap parameters for Rb and Cs atoms in a 3D optical lattice required to create single occupancy per site Mott phase for both the species in the same regions of the trap. We thus identify the ne tuning of the external magnetic held near Rb-Cs Feshbach resonance required to achieve highest probability for creating single Rb-Cs Feshbach molecules in the system. We have used the Falicov-Kimball model to describe the relevant system and strong-coupling expansions about the mean- held solution to calculate the density pro les for both species and efficiency for molecule formation, determined by overlapping regions of single occupancy for both Rb and Cs, up to second order in the expansion. We also calculate the entropy per particle which serves as an estimation of the temperature in the experimental system

Les méthodes de Monte Carlo par chaines de Markov MCMC sont des outils mathématiques utilisés pour simuler des mesures de probabilités π définies sur des espaces de grandes dimensions. Une des questions les plus importantes dans ce contexte est de savoir à quelle vitesse converge la chaine de Markov P vers la mesure invariante π. Pour mesurer la vitesse de convergence de la chaine de Markov P vers sa mesure invariante π nous utilisons la distance de la variation totale. Il est bien connu que la vitesse de convergence d’une chaine de Markov réversible P dépend de la deuxième plus grande valeur propre en valeur absolue de la matrice P notée β!. Une partie importante dans l’estimation de β! consiste à estimer la deuxième plus grande valeur propre de la matrice P, qui est notée β1. Diaconis et Stroock (1991) ont introduit une méthode basée sur l’inégalité de Poincaré pour estimer β1 pour le cas général des chaines de Markov réversibles avec un nombre fini d'état. Dans cette thèse, nous utilisons la méthode de Shiu et Chen (2015) pour étudier le cas de l'algorithme de l'échantillonneur de Gibbs pour le modèle d'Ising unidimensionnel avec trois états ou plus appelé aussi modèle de Potts. Puis, nous généralisons le résultat de Shiu et Chen au cas du modèle d’Ising deux- dimensionnel avec deux états. Les résultats obtenus minorent ceux introduits par Ingrassia (1994). Puis nous avons pensé à perturber l'échantillonneur de Gibbs afin d’améliorer sa vitesse de convergence vers l'équilibre. / Monte Carlo Markov chain methods MCMC are mathematical tools used to simulate probability measures π defined on state spaces of high dimensions. The speed of convergence of this Markov chain X to its invariant state π is a natural question to study in this context.To measure the convergence rate of a Markov chain we use the total variation distance. It is well known that the convergence rate of a reversible Markov chain depends on its second largest eigenvalue in absolute value denoted by β!. An important part in the estimation of β! is the estimation of the second largest eigenvalue which is denoted by β1.Diaconis and Stroock (1991) introduced a method based on Poincaré inequality to obtain a bound for β1 for general finite state reversible Markov chains.In this thesis we use the Chen and Shiu approach to study the case of the Gibbs sampler for the 1−D Ising model with three and more states which is also called Potts model. Then, we generalize the result of Shiu and Chen (2015) to the case of the 2−D Ising model with two states.The results we obtain improve the ones obtained by Ingrassia (1994). Then, we introduce some method to disrupt the Gibbs sampler in order to improve its convergence rate to equilibrium.

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