<p> In premixed turbulent combustion, the normal speed of propagation of the flame front is enhanced by the turbulent velocity field. This project will focus on the method of computing the normal speed of propagation of the flame front in the Majda-Souganidis model of turbulent combustion. Solving this problem involves computing the eigenvalue of a nonlinear cell problem. Discussed in this thesis is a new, simple and direct numerical method for approximating the eigenvalue, also called the effective Hamiltonian.</p> / Thesis / Master of Science (MSc)

Les propriétés des matériaux ferroélectriques proviennent principalement de l’influencedes conditions aux limites et des déformations sur la polarisation. Cette influence est encoreplus grande à de petites échelles ou des structures particulières de la polarisation apparaissent,comme les vortex dans les cubes quantiques ou des structures en rayures dans lescouches minces. Pour le calcul, à très basses échelles, de telles structures de polarisation, lesHamiltonien effectifs, basés sur les calculs ab-initio sont les plus utilisés. Parallèlement Lesmodèles continus sont préconisés à plus grandes échelles. Néanmoins, il n’existe pas de lienentre ces deux modèles. Le but de cette thèse est alors de construire une approche permettantde relier ces deux modèles et par cela même ces différentes échelles.Notre modèle se base sur un Hamiltonien effectif écrit pour le titanate de baryum enfonction de la polarisation et des déformations. Cet Hamiltonien est reformulé de façon àdécrire un milieu continu. Les difficultés de cette reformulation proviennent des interactionsnon locales. Le résultat est alors un système d’équations aux dérivées partielles, décrivantl’équilibre et les conditions aux limites. La température est ensuite introduite de façon effectivedans les coefficients de ces équations. Notre modèle ressemble fortement aux modèlesde Landau.Une telle approche est appliquée dans les cubes quantiques et les couches minces óu l’organisationdes domaines dépend de la taille. Les résultats montrent l’implication de la méthodedes éléments finis sur la précision. La formation de vortex dans les cubes quantiquesest bien reproduite. L’agencement en domaines de polarisation alternée dans les couchesminces est elle aussi bien reproduite pour les couches minces. De plus en augmentant l’épaisseurde ces couches minces, la périodicité de cet agencement alterné est modifié, comportementdécrit par la loi de Kittel qui est ici calculée et comparée aux résultats expérimentaux. / Physicals properties of ferroelectric materials mainly arise from the fact that the polarizationis strongly influenced by strain and electrical boundary conditions, which may changeits orientation and magnitude. At small scales, this influence is even stronger and unusualdomain structures are produced like vortices in quantum dots or stripes in thin films. For thecalculation of domain structures, at small scales, first-principle-based effective Hamiltonianare widely used whereas at higher scales, continuum models are predominants. Nevertheless,in between there is no computational method connecting both scales. Therefore„ thegoal of this dissertation is to develop and build new approaches in order to bridge these twoseparated scales.Our model stems for classical effective Hamiltonian, written for barium titanate as afunction of the polarization and strain. This Hamiltonian is then formulated in order tocorrespond to a continuous description. Difficulties arise from non local interactions. In theend, the Hamiltonian is transformed into a set of partial differential equations describing theequilibrium and the boundary conditions. The temperature is then introduced in such a waythat makes evolve the coefficients of those sets of equations. We therefore reconstructed aLandu-like model.Such approach can be applied in quantum dots and thin films where the domain organizationdepend on the size. The results show how to apply finite element in order to obtainpatterns of polarizations with the wanted precision. The vortices shapes of domain patternin quantum dots is well reproduced. The stripes-like polarization pattern is also well reproducedin thin films. Besides expanding thickness of those films change the periodicity ofthose stripes, behaviour described by the Kittel law. This law is calculated and compared tomeasurements.

Le défis principal de l'industrie de la micro électronique est de créer d'augmenter la capacité de stockage mais aussi la vitesse des ordinateurs. Pour atteindre cette objectif, les composants électroniques doivent être miniaturisés à l'échelle du nanomètre. À cette échelle, les propriétés de la matière sont encore mal connues.Les matériaux les plus prometteurs dans cette recherche sont les multiferroïques où l'ordre magnétique et l'ordre ferroélectrique sont couplés. Ils pourraient amener des composants électroniques plus rapide et moins consommateur d'énergie dans des composants tels que les Random Access Memory. Ce travail traite de l'étude d'un multiferroïque typique BiFeO3 (BFO) en se concentrant sur les couplages entre les ordres magnétiques, ferroélectriques et le contrainte dans des systèmes de taille nanométrique / A major challenge in microelectronics is the increase of data storage as well as processors performancies. Unfortunatelly, this challenge involves a drastic reduction of size of the fundamental device of a computer down to the nano scale. At this scale, properties of matter are still not fully understood. One of the key materials to reach this challenge are multiferroics where the magnetism and the ferroelectricity can interact leading to low consuming and fast Random Access Memories. This work deals with the study of famous multiferroics BiFeO3 (BFO) focussing on the coupling between magnetic ordering, ferroelectric ordering and strain as the dimensionality of the system is reduced to several nanometers

In this thesis Moore and Yeo's proposed mapping of the structural glass to the Ising spin glass in a random field is presented. In contrast to Random First Order Theory and Mode Coupling Theory, this mapping predicts that there should be no glass transition at finite temperature. However, a growing correlation length is predicted from the size of rearranging regions in the supercooled liquid, and from this a growing structural relaxation time is predicted. Also presented is a study of the propensity of binary fluids (i.e. fluids containing particles of two sizes) to phase separate into regions dominated by one type of particle only. Binary fluids like this are commonly used as model glass formers and the study shows that this phase separation behaviour is something that must be taken into account.The mapping relies on the use of replica theory and is therefore very opaque. Here a model is presented that may be mapped directly to a system of spins, and also prevents the process of phase separation from occurring in binary fluids. The system of spins produced in the mapping is then analysed through the use of an effective Hamiltonian, which is in the universality class of the Ising spin glass in a random field. The behaviour of the correlation length depends on the spin-spin coupling J and the strength of the random field h. The variation of these with packing fraction and temperature T is studied for a simple model, and the results extended to the full system. Finally a prediction is made for the critical exponents governing the correlation length and structural relaxation time.

Mean-field games (MFGs) study the behavior of rational and indistinguishable agents in a large population. Agents seek to minimize their cost based upon statis- tical information on the population’s distribution. In this dissertation, we study the homogenization of a stationary first-order MFG and seek to find a numerical method to solve the homogenized problem. More precisely, we characterize the asymptotic behavior of a first-order stationary MFG with a periodically oscillating potential. Our main tool is the two-scale convergence. Using this convergence, we rigorously derive the two-scale homogenized and the homogenized MFG problems. Moreover, we prove existence and uniqueness of the solution to these limit problems. Next, we notice that the homogenized problem resembles the problem involving effective Hamiltoni- ans and Mather measures, which arise in several problems, including homogenization of Hamilton–Jacobi equations, nonlinear control systems, and Aubry–Mather theory. Thus, we develop algorithms to solve the homogenized problem, the effective Hamil- tonians, and Mather measures. To do that, we construct the Hessian Riemannian flow. We prove the convergence of the Hessian Riemannian flow in the continuous setting. For the discrete case, we give both the existence and the convergence of the Hessian Riemannian flow. In addition, we explore a variant of Newton’s method that greatly improves the performance of the Hessian Riemannian flow. In our numerical experiments, we see that our algorithms preserve the non-negativity of Mather mea- sures and are more stable than related methods in problems that are close to singular. Furthermore, our method also provides a way to approximate stationary MFGs.

We pursue topics in optics that follow three major themes; time averaged dynamics with the associated Effective Hamiltonian theory, quantification and transformation of polarization, and periodicity within quantum circuits. Within the first theme, we develop a technique for finding the dynamical evolution in time of a time averaged density matrix. The result is an equation of evolution that includes an Effective Hamiltonian, as well as decoherence terms that sometimes manifest in a Lindblad-like form. We also apply the theory to examples of the AC Stark Shift and Three-Level Raman Transitions. In the theme of polarization, the most general physical transformation on the polarization state has been represented as an ensemble of Jones matrix transformations, equivalent to a completely positive map on the polarization matrix. This has been directly assumed without proof by most authors. We follow a novel approach to derive this expression from simple physical principles, basic coherence optics and the matrix theory of positive maps. Addressing polarization measurement, we first establish the equivalence of classical polarization and quantum purity, which leads to the identical structure of the Poincar\' and Bloch spheres. We analyze and compare various measures of polarization / purity for general dimensionality proposed in the literature, with a focus on the three dimensional case. % entanglement? In pursuit of the final theme of periodic quantum circuits, we introduce a procedure that synthesizes the circuit for the simplest periodic function that is one-to-one within a single period, of a given period p. Applying this procedure, we synthesize these circuits for p up to five bits. We conjecture that such a circuit will need at most n Toffoli gates, where p is an n-bit number. Moreover, we apply our circuit synthesis to compiled versions of Shor's algorithm, showing that it can create more efficient circuits than ones previously proposed. We provide some new compiled circuits for experimentalists to use in the near future. A layer of "classical compilation" is pointed out as a method to further simplify circuits. Periodic and compiled circuits should be helpful for creating experimental milestones, and for the purposes of validation.

We pursue topics in optics that follow three major themes; time averaged dynamics with the associated Effective Hamiltonian theory, quantification and transformation of polarization, and periodicity within quantum circuits. Within the first theme, we develop a technique for finding the dynamical evolution in time of a time averaged density matrix. The result is an equation of evolution that includes an Effective Hamiltonian, as well as decoherence terms that sometimes manifest in a Lindblad-like form. We also apply the theory to examples of the AC Stark Shift and Three-Level Raman Transitions. In the theme of polarization, the most general physical transformation on the polarization state has been represented as an ensemble of Jones matrix transformations, equivalent to a completely positive map on the polarization matrix. This has been directly assumed without proof by most authors. We follow a novel approach to derive this expression from simple physical principles, basic coherence optics and the matrix theory of positive maps. Addressing polarization measurement, we first establish the equivalence of classical polarization and quantum purity, which leads to the identical structure of the Poincar\' and Bloch spheres. We analyze and compare various measures of polarization / purity for general dimensionality proposed in the literature, with a focus on the three dimensional case. % entanglement? In pursuit of the final theme of periodic quantum circuits, we introduce a procedure that synthesizes the circuit for the simplest periodic function that is one-to-one within a single period, of a given period p. Applying this procedure, we synthesize these circuits for p up to five bits. We conjecture that such a circuit will need at most n Toffoli gates, where p is an n-bit number. Moreover, we apply our circuit synthesis to compiled versions of Shor's algorithm, showing that it can create more efficient circuits than ones previously proposed. We provide some new compiled circuits for experimentalists to use in the near future. A layer of "classical compilation" is pointed out as a method to further simplify circuits. Periodic and compiled circuits should be helpful for creating experimental milestones, and for the purposes of validation.

The LiHo_xY_{1-x}F_4 magnetic material in a transverse magnetic field B_x perpendicular to the Ising spin direction has long been used to study tunable quantum phase transitions in pure and random disordered systems. We first present analytical and numerical evidences for the validity of an effective spin-1/2 approach to the description of a general dipolar spin glass model with strong uniaxial Ising anisotropy and subject to weak B_x. We relate this toy model to the LiHo_xY_{1-x}F_4 transverse field Ising material. We show that an effective spin-1/2 model is able to capture both the qualitative and quantitative aspects of the physics at small B_x. After confirming the validity of the effective spin-1/2 approach, we show that the field-induced magnetization along the x direction, combined with the local random dilution-induced destruction of crystalline mirror symmetries generates, via the predominant dipolar interactions between Ho^{3+} ions, random fields along the Ising z direction. This identifies LiHo_xY_{1-x}F_4 in B_x as a new random field Ising system. We show that the random fields explain the smearing of the nonlinear susceptibility at the spin glass transition with increasing B_x. In this thesis, we also investigate the phase diagram of non-diluted LiHoF_4 in the presence of B_x, by performing Monte-Carlo simulations. A previous quantum Monte Carlo (QMC) simulation found that even for small B_x where quantum fluctuations are small, close to the classical critical point, there is a discrepancy between experiment and the QMC results. We revisit this problem, focusing on weak B_x close to the classical T_c, using an alternative approach. For small B_x, by applying a so-called cumulant expansion, the quantum fluctuations around the classical T_c are taken into account perturbatively. We derived an effective perturbative classical Hamiltonian, on which MC simulations are performed. With this method we investigate different proposed sources of uncertainty which can affect the numerical results. We fully reproduce the previous QMC results at small B_x. Unfortunately, we find that none of the modifications to the microscopic Hamiltonian that we explore are able to provide a B_x-T phase diagram compatible with the experiments in the small semi-classical B_x regime.

The LiHo_xY_{1-x}F_4 magnetic material in a transverse magnetic field B_x perpendicular to the Ising spin direction has long been used to study tunable quantum phase transitions in pure and random disordered systems. We first present analytical and numerical evidences for the validity of an effective spin-1/2 approach to the description of a general dipolar spin glass model with strong uniaxial Ising anisotropy and subject to weak B_x. We relate this toy model to the LiHo_xY_{1-x}F_4 transverse field Ising material. We show that an effective spin-1/2 model is able to capture both the qualitative and quantitative aspects of the physics at small B_x. After confirming the validity of the effective spin-1/2 approach, we show that the field-induced magnetization along the x direction, combined with the local random dilution-induced destruction of crystalline mirror symmetries generates, via the predominant dipolar interactions between Ho^{3+} ions, random fields along the Ising z direction. This identifies LiHo_xY_{1-x}F_4 in B_x as a new random field Ising system. We show that the random fields explain the smearing of the nonlinear susceptibility at the spin glass transition with increasing B_x. In this thesis, we also investigate the phase diagram of non-diluted LiHoF_4 in the presence of B_x, by performing Monte-Carlo simulations. A previous quantum Monte Carlo (QMC) simulation found that even for small B_x where quantum fluctuations are small, close to the classical critical point, there is a discrepancy between experiment and the QMC results. We revisit this problem, focusing on weak B_x close to the classical T_c, using an alternative approach. For small B_x, by applying a so-called cumulant expansion, the quantum fluctuations around the classical T_c are taken into account perturbatively. We derived an effective perturbative classical Hamiltonian, on which MC simulations are performed. With this method we investigate different proposed sources of uncertainty which can affect the numerical results. We fully reproduce the previous QMC results at small B_x. Unfortunately, we find that none of the modifications to the microscopic Hamiltonian that we explore are able to provide a B_x-T phase diagram compatible with the experiments in the small semi-classical B_x regime.

We present in this thesis a series of studies on the physical properties of some one dimensional systems. In particular we study the low energy properties of various spin chains and a junction of Luttinger wires. For spin chains we specifically look at the role of perturbations like frustrating interactions and dimerisation in a nearest neighbour chain and the formation of magnetisation plateaus in two kinds of models; one purely theoretical and the other motivated by experiments. In our second subject of interest we study using a renormalisation group analysis the effect of spin dependent scattering at a junction of Luttinger wires. We look at the physical effects caused by the interplay of electronic interactions in the wires and the scattering processes at the junction. The thesis begins with an introductory chapter which gives a brief glimpse of the ideas and techniques used in the specific problems that we have worked on. Our work on these problems is then described in detail in chapters 25. We now present a brief summary of each of those chapters. In the second chapter we look at the ground state phase diagram of the mixed-spin sawtooth chain, i.e a system where the spins along the baseline are allowed to be different from the spins on the vertices. The spins S1 along the baseline interact with a coupling strength J1(> 0). The coupling of the spins on the vertex (S2) to the baseline spins has a strength J2. We study the phase diagram as a function of J2/J1 [1]. The model exhibits a rich variety of phases which we study using spinwave theory, exact diagonalisation and a semi-numerical perturbation theory leading to an effective Hamiltonian. The spinwave theory predicts a transition from a spiral state to a ferrimagnetic state at J2S2/2J1S1 = 1 as J2/J1 is increased. The spectrum has two branches one of which is gapless and dispersionless (at the linear order) in the spiral phase. This arises because of the infinite degeneracy of classical ground states in that phase. Numerically, we study the system using exact diagonalisation of up to 12 unit cells and S1 = 1 and S2 =1/2. We look at the variation of ground state energy, gap to the lowest excitations, and the relevant spin correlation functions in the model. This unearths a richer phase diagram than the spinwave calculation. Apart from revealing a possibility of the presence of more than one kind of spiral phases, numerical results tell us about a very interesting phase for small J2. The spin correlation function (for the spin1/2s) in this region have a property that the nextnearest-neighbour correlations are much larger than the nearest neighbour correlations. We call this phase the NNNAFM (nextnearest neighbour antiferromagnet) phase and provide an understanding of this phase by deriving an effective Hamiltonian between the spin1/2s. We also show the existence of macroscopic magnetisation jumps in the model when one looks at the system close to saturation fields. The third chapter is concerned with the formation of magnetisation plateaus in two different spin models. We show how in one model the plateaus arise because of the competition between two coupling constants, and in the other because of purely geometrical effects. In the first problem we propose [2] a class of spin Hamiltonians which include as special cases several known systems. The class of models is defined on a bipartite lattice in arbitrary dimensions and for any spin. The simplest manifestation of such models in one dimension corresponds to a ladder system with diagonal couplings (which are of the same strength as the leg couplings). The physical properties of the model are determined by the combined effects of the competition between the ”rung” coupling (J’ )and the ”leg/diagonal” coupling (J ) and the magnetic field. We show that our model can be solved exactly in a substantial region of the parameter space (J’ > 2J ) and we demonstrate the existence of magnetisation plateaus in the solvable regime. Also, by making reasonable assumptions about the spectrum in the region where we cannot solve the model exactly, we prove the existence of first order phase transitions on a plateau where the sublattice magnetisations change abruptly. We numerically investigate the ladder system mentioned above (for spin1) to confirm all our analytical predictions and present a phase diagram in the J’/J - B plane, quite a few of whose features we expect to be generically valid for all higher spins. In the second problem concerning plateaus (also discussed in chapter 3) we study the properties of a compound synthesised experimentally [3]. The essential feature of the structure of this compound which gives rise to its physical properties is the presence of two kinds of spin1/2 objects alternating with each other on a helix. One kind has an axis of anisotropy at an inclination to the helical axis (which essentially makes it an Ising spin) whereas the other is an isotropic spin1/2 object. These two spin1/2 objects interact with each other but not with their own kind. Experimentally, it was observed that in a magnetic field this material exhibits magnetisation plateaus one of which is at 1/3rd of the saturation magnetisation value. These plateaus appear when the field is along the direction of the helical axis but disappear when the field is perpendicular to that axis. The model being used for the material prior to our work could not explain the existence of these plateaus. In our work we propose a simple modification in the model Hamiltonian which is able to qualitatively explain the presence of the plateaus. We show that the existence of the plateaus can be explained using a periodic variation of the angles of inclination of the easy axes of the anisotropic spins. The experimental temperature and the fields are much lower than the magnetic coupling strength. Because of this quite a lot of the properties of the system can be studied analytically using transfer matrix methods for an effective theory involving only the anisotropic spins. Apart from the plateaus we study using this modified model other physical quantities like the specific heat, susceptibility and the entropy. We demonstrate the existence of finite entropy per spin at low temperatures for some values of the magnetic field. In chapter 4 we investigate the longstanding problem of locating the gapless points of a dimerised spin chain as the strength of dimerisation is varied. It is known that generalising Haldane’s field theoretic analysis to dimerised spin chains correctly predicts the number of the gapless points but not the exact locations (which have determined numerically for a few low values of spins). We investigate the problem of locating those points using a dimerised spin chain Hamiltonian with a ”twisted” boundary condition [4]. For a periodic chain, this ”twist” consists simply of a local rotation about the zaxis which renders the xx and yy terms on one bond negative. Such a boundary condition has been used earlier for numerical work whereby one can find the gapless points by studying the crossing points of ground states of finite chains (with the above twist) in different parity sectors (parity sectors are defined by the reflection symmetry about the twisted bond). We study the twisted Hamiltonian using two analytical methods. The modified boundary condition reduces the degeneracy of classical ground states of the chain and we get only two N´eel states as classical ground states. We use this property to identify the gapless points as points where the tunneling amplitude between these two ground states goes to zero. While one of our calculations just reproduces the results of previous field theoretic treatments, our second analytical treatment gives a direct expression for the gapless points as roots of a polynomial equation in the dimerisation parameter. This approach is found to be more accurate. We compare the two methods with the numerical method mentioned above and present results for various spin values. In the final chapter we present a study of the physics of a junction of Luttinger wires (quantum wires) with both scalar and spin scattering at the junction ([5],[6]). Earlier studies have investigated special cases of this system. The systems studied were two wire junctions with either a fully transmitting scattering matrix or one corresponding to disconnected wires. We extend the study to a junction of N wires with an arbitrary scattering matrix and a spin impurity at the junction. We study the RG flows of the Kondo coupling of the impurity spin to the electrons treating the electronic interactions and the Kondo coupling perturbatively. We analyse the various fixed points for the specific case of three wires. We find a general tendency to flow towards strong coupling when all the matrix elements of the Kondo coupling are positive at small length scales. We analyse one of the strong coupling fixed points, namely that of the maximally transmitting scattering matrix, using a 1/J perturbation theory and we find at large length scales a fixed point of disconnected wires with a vanishing Kondo coupling. In this way we obtain a picture of the RG at both short and long length scales. Also, we analyse all the fixed points using lattice models to gain an understanding of the RG flows in terms of specific couplings on the lattice. Finally, we use to bosonisation to study one particular case of scattering (the disconnected wires) in the presence of strong interactions and find that sufficiently strong interactions can stabilise a multichannel fixed point which is unstable in the weak interaction limit.