La technique des répliques est une technique formidable prenant ses origines de la physique statistique, comme un moyen de calculer l'espérance du logarithme de la constante de normalisation d'une distribution de probabilité à haute dimension. Dans le jargon de physique, cette quantité est connue sous le nom de l’énergie libre, et toutes sortes de quantités utiles, telle que l’entropie, peuvent être obtenue de là par des dérivées. Cependant, ceci est un problème NP-difficile, qu’une bonne partie de statistique computationelle essaye de résoudre, et qui apparaît partout; de la théorie des codes, à la statistique en hautes dimensions, en passant par les problèmes de satisfaction de contraintes. Dans chaque cas, la méthode des répliques, et son extension par (Parisi et al., 1987), se sont prouvées fortes utiles pour illuminer quelques aspects concernant la corrélation des variables de la distribution de Gibbs et la nature fortement nonconvexe de son logarithme negatif. Algorithmiquement, il existe deux principales méthodologies adressant la difficulté de calcul que pose la constante de normalisation: a). Le point de vue statique: dans cette approche, on reformule le problème en tant que graphe dont les nœuds correspondent aux variables individuelles de la distribution de Gibbs, et dont les arêtes reflètent les dépendances entre celles-ci. Quand le graphe en question est localement un arbre, les procédures de message-passing sont garanties d’approximer arbitrairement bien les probabilités marginales de la distribution de Gibbs et de manière équivalente d'approximer la constante de normalisation. Les prédictions de la physique concernant la disparition des corrélations à longues portées se traduise donc, par le fait que le graphe soit localement un arbre, ainsi permettant l’utilisation des algorithmes locaux de passage de messages. Ceci va être le sujet du chapitre 4. b). Le point de vue dynamique: dans une direction orthogonale, on peut contourner le problème que pose le calcul de la constante de normalisation, en définissant une chaîne de Markov le long de laquelle, l’échantillonnage converge à celui selon la distribution de Gibbs, tel qu’après un certain nombre d’itérations (sous le nom de temps de relaxation), les échantillons sont garanties d’être approximativement générés selon elle. Afin de discuter des conditions dans lesquelles chacune de ces approches échoue, il est très utile d’être familier avec la méthode de replica symmetry breaking de Parisi. Cependant, les calculs nécessaires sont assez compliqués, et requièrent des notions qui sont typiquemment étrangères à ceux sans un entrainement en physique statistique. Ce mémoire a principalement deux objectifs : i) de fournir une introduction a la théorie des répliques, ses prédictions, et ses conséquences algorithmiques pour les problèmes de satisfaction de constraintes, et ii) de donner un survol des méthodes les plus récentes adressant la transition de phase, prédite par la méthode des répliques, dans le cas du problème k−SAT, à partir du point de vu statique et dynamique, et finir en proposant un nouvel algorithme qui prend en considération la transition de phase en question. / The replica trick is a powerful analytic technique originating from statistical physics as an attempt to compute the expectation of the logarithm of the normalization constant of a high dimensional probability distribution known as the Gibbs measure. In physics jargon this quantity is known as the free energy, and all kinds of useful quantities, such as the entropy, can be obtained from it using simple derivatives. The computation of this normalization constant is however an NP-hard problem that a large part of computational statistics attempts to deal with, and which shows up everywhere from coding theory, to high dimensional statistics, compressed sensing, protein folding analysis and constraint satisfaction problems. In each of these cases, the replica trick, and its extension by (Parisi et al., 1987), have proven incredibly successful at shedding light on keys aspects relating to the correlation structure of the Gibbs measure and the highly non-convex nature of − log(the Gibbs measure()). Algorithmic speaking, there exists two main methodologies addressing the intractability of the normalization constant: a) Statics: in this approach, one casts the system as a graphical model whose vertices represent individual variables, and whose edges reflect the dependencies between them. When the underlying graph is locally tree-like, local messagepassing procedures are guaranteed to yield near-exact marginal probabilities or equivalently compute Z. The physics predictions of vanishing long range correlation in the Gibbs measure, then translate into the associated graph being locally tree-like, hence permitting the use message passing procedures. This will be the focus of chapter 4. b) Dynamics: in an orthogonal direction, we can altogether bypass the issue of computing the normalization constant, by defining a Markov chain along which sampling converges to the Gibbs measure, such that after a number of iterations known as the relaxation-time, samples are guaranteed to be approximately sampled according to the Gibbs measure. To get into the conditions in which each of the two approaches is likely to fail (strong long range correlation, high energy barriers, etc..), it is very helpful to be familiar with the so-called replica symmetry breaking picture of Parisi. The computations involved are however quite involved, and come with a number of prescriptions and prerequisite notions (s.a. large deviation principles, saddle-point approximations) that are typically foreign to those without a statistical physics background. The purpose of this thesis is then twofold: i) to provide a self-contained introduction to replica theory, its predictions, and its algorithmic implications for constraint satisfaction problems, and ii) to give an account of state of the art methods in addressing the predicted phase transitions in the case of k−SAT, from both the statics and dynamics points of view, and propose a new algorithm takes takes these into consideration.

Les matériaux de Dirac constituent une classe intéressante de systèmes pouvant subir une transition de phase quantique à température nulle, lorsqu’un paramètre non-thermique atteint un point critique quantique. À l’approche d’un tel point, les observables physiques sont affectées par les importantes fluctuations thermiques et quantiques. Dans ce mémoire, on utilise des techniques de théorie conforme des champs afin d’étudier le tenseur de conductivité électrique dans des théories en 2 + 1 dimensions contenant des fermions de Dirac près d’un point critique quantique. À basse énergie, ces dernières décrivent de façon adéquate de nombreux matériaux de Dirac ainsi que leur transition de phase quantique. La conductivité est étudiée dans le régime des hautes fréquences, à température non-nulle et lorsque le paramètre non-thermique est près de sa valeur critique. Dans ce projet, l’emphase est mise sur les points critiques quantiques invariants sous la parité et le renversement du temps. Dans ce cas, l’expansion de produit d’opérateurs (Operator product expansion en anglais) ainsi que la théorie des perturbations conforme permettent d’obtenir une expression générale pour l’expansion à grandes fréquences des conductivités longitudinales et transverses (de Hall) lorsque le point critique quantique est déformé par un opérateur scalaire relevant. Grâce à ces dernières, nous sommes en mesure de déduire des règles de somme exactes pour ces deux quantités. À titre d’exemple, nos résultats généraux sont appliqués dans le cadre du modèle interagissant de Gross-Neveu, où nous obtenons l’expansion des deux conductivités ainsi que les règles de somme pour un nombre de saveurs de fermions de Dirac N arbitraire. Ces mêmes expressions sont ensuite obtenues par un calcul explicite à N = infini, permettant la comparaison avec les résultats pour un N quelconque. Par la suite, des résultats généraux similaires sont obtenus dans le cas où le point critique quantique est déformé par un opérateur pseudoscalaire relevant. Ces derniers sont finalement appliqués à une théorie de fermions de Dirac libres perturbée par un terme de masse. / Dirac materials constitute an interesting class of systems that can undergo a quantum phase transition at zero temperature, when a non-thermal parameter reaches a quantum critical point. As we approach such a point, physical observables are altered by the important thermal and quantum fluctuations. In this thesis, conformal field theory techniques are used to study the electrical conductivity tensor in theories with Dirac fermions in 2+1 dimensions close to a quantum critical point. At low energies, these adequately describe various Dirac materials as well as their quantum phase transition. In this project, we focus on theories that have a quantum critical point invariant under parity and time-reversal. In this case, the operator product expansion and conformal perturbation theory allow to obtain a general expression for the large frequency expansion of the longitudinal and transverse (Hall) conductivities when the quantum critical point is deformed by a relevant scalar operator. Using these, we are able to deduce exact sum rules for both quantities. As an example, our general results are applied to the Gross-Neveu model, where we obtain the large frequency expansion for both conductivities and the associated sum rules for an arbitrary number of Dirac fermion flavors N. The same expressions are then obtained by an explicit calculation at N = infinity, allowing to compare with our results for any N. Afterwards, analogous general results are obtained for theories where the quantum critical point is deformed by a relevant pseudoscalar. These are finally applied to a theory of massless free Dirac fermions perturbed by a mass term.

The present thesis is devoted to the investigation of modulated structures as well as the direct measurement of magnetocaloric effect (MCE) in Ni-Mn based magnetic shape memory (MSM) Heusler compounds in pulsed magnetic fields after analyzing isothermal entropy data taken in static magnetic fields. The emphasis is on the modulated structure of MSM Heusler compounds because of lower twinning stress which facilitates the easy transformation from austenite to martensite structure. Synchrotron x-ray powder diffraction (SXRPD) was carried out to study the modulated structure and NPD for antisite disorder as Ni and Mn have easily the same atomic scattering factor. Direct measurement of the adiabatic temperature change ΔTad was done in pulsed magnetic fields, because of fast response of ~10 to 100 ms to the sample temperature on magnetic field, providing adiabatic conditions. It also gives an opportunity of very high magnetic fields up to 70 T because of short pulse duration during the measurement. The modulated structure has been studied for the off-stoichiometric Ni2Mn1.4In0.6 and Ni1.9Pt0.1MnGa MSM Heusler compounds from SXRPD and NPD. Ni2Mn1.4In0.6 exhibits martensitic transition at TM ~ 295 K and Curie temperature TC ~ 315 K. Rietveld refinement reveals uniform atomic displacement in the modulated structure of martensite phase and the absence of premartensite phase and phason broadening of the satellite peaks which was further confirmed by HRTEM study. Therefore, the structural modulation in Ni2Mn1.4In0.6 can be successfully explained in term of the adaptive phase model. Whereas, Ni1.9Pt0.1MnGa shows the premartensite phase in addition to the martensite and austenite phases and follows the soft phonon model. The temperature dependent ac-susceptibility shows the change in slope at different temperatures 365, 265, 230 and 220 K corresponding to the Curie temperature TC, first premartensite T1, second premartensite T2 and martensite temperature TM, respectively. Temperature-dependent high resolution SXRPD data analysis shows first, a nearly 3M modulated premartensite phase with an average cubic-like feature i.e. negligible Bain distortion of the elementary L21 unit cell results from the austenite phase. This phase then undergoes an isostructural phase transition 3M like premartensite phase with robust Bain distortion in the temperature range from 220 to 195 K. Below 195 K, the martensite phase appears which results from the larger Bain-distorted premartensite phase. In this work, the magnetocaloric properties of Ni2.2Mn0.8Ga and Ni1.8Mn1.8In0.4 magnetic shape memory (MSM) Heusler compounds were studied. Ni2.2Mn0.8Ga exhibits the reversible conventional MCE, measured from isothermal entropy change ΔSM and adiabatic temperature change ΔTad because of the geometric compatibility condition (GCC) for cubic austenite phase to tetragonal martensite phase as a consequence of low thermal hysteresis of the martensite phase transition. The reversible MCE has been confirmed by applying more than one pulse in the hysteresis region at 317 K. Ni1.8Mn1.8In0.4 possess improved reversible behavior of inverse MCE due to the closely satisfying of GCC from cubic austenite to modulated monoclinic martensite structure. The maximum value of ΔSM has been found to the same for both heating and cooling curves measured from isothermal magnetization M(T) curves until a magnetic field of 5 T. The adiabatic temperature change ΔTad results in a value of -10 K by applying a magnetic field of 20 T in a pulsed magnetic field. Furthermore, reversible magnetostriction of 0.3% was observed near the first-order martensite phase transition temperatures 265, 270 and 280 K. A reduction of thermal hysteresis has been found in MSM Heusler compounds Ni2Mn1.4In0.6 and Ni1.8Co0.2Mn1.4In0.6 with the application of hydrostatic pressure followed by GCC from pressure dependent x-ray diffraction in both austenite and martensite phase. By increasing pressure, the lattice parameters of both phases change in such a way that they increasingly satisfy the GCC. The approach of GCC for different kind of martensite structures (tetragonal, orthorhombic and monoclinic) will help to design new MSM Heusler compounds taking advantage of first-order martensite phase transition.