Thesis advisor: Ziqiang . Wang / The transition between many-body localized states and the delocalized thermal states is an eigenstate phase transition at finite energy density outside the scope of conventional quantum statistical mechanics. We apply support vector machine (SVM) to study the phase transition between many-body localized and thermal phases in a disordered quantum Ising chain in a transverse external field. The many-body eigenstate energy E is bounded by a bandwidth W=Eₘₐₓ-Eₘᵢₙ. The transition takes place on a phase diagram spanned by the energy density ϵ=2(Eₘₐₓ-Eₘᵢₙ)/W and the disorder strength ẟJ of the spin interaction uniformly distributed within [-ẟJ, ẟJ], formally parallel to the mobility edge in Anderson localization. In our study we use the labeled probability density of eigenstate wavefunctions belonging to the deeply localized and thermal regimes at two different energy densities (ϵ's) as the training set, i.e., providing labeled data at four corners of the phase diagram. Then we employ the trained SVM to predict the whole phase diagram. The obtained phase boundary qualitatively agrees with previous work using entanglement entropy to characterize these two phases. We further analyze the decision function of the SVM to interpret its physical meaning and find that it is analogous to the inverse participation ratio in configuration space. Our findings demonstrate the ability of the SVM to capture potential quantities that may characterize the many-body localization phase transition. To further investigate the properties of the transition, we study the behavior of the entanglement entropy of a subsystem of size L_A in a system of size L > L_A near the critical regime of the many-body localization transition. The many-body eigenstates are obtained by exact diagonalization of a disordered quantum spin chain under twisted boundary conditions to reduce the finite-size effect. We present a scaling theory based on the assumption that the transition is continuous and use the subsystem size L_A/ξ as the scaling variable, where ξ is the correlation length. We show that this scaling theory provides an effective description of the critical behavior and that the entanglement entropy follows the thermal volume law at the transition point. We extract the critical exponent governing the divergence of ξ upon approaching the transition point. We again study the participation entropy in the spin-basis of the domain wall excitations and show that the transition point and the critical exponent agree with those obtained from finite size scaling of the entanglement entropy. Our findings suggest that the many-body localization transition in this model is continuous and describable as a localization transition in the many-body configuration space. Besides the many-body localization transition driven by disorder, We also study the Coulomb repulsion and temperature driving phase transitions. We apply a finite-temperature Gutzwiller projection to two-dimensional Hubbard model by constructing a "Gutzwiller-type" density matrix operator to approximate the real interacting density matrix, which provides the upper bound of free energy of the system. We firstly investigate half filled Hubbard model without magnetism and obtain the phase diagram. The transition line is of first order at finite temperature, ending at 2 second order points, which shares qualitative agreement with dynamic mean field results. We derive the analytic form of the free energy and therefor the equation of states, which benefits the understanding of the different phases. We later extend our approach to take anti-ferromagnetic order into account. We determine the Neel temperature and explore its interesting behavior when varying the Coulomb repulsion. / Thesis (PhD) — Boston College, 2019. / Submitted to: Boston College. Graduate School of Arts and Sciences. / Discipline: Physics.

L’exploration de systèmes quantiques à N corps fortement corrélés représente l’un des domaines de recherche les plus stimulants de la physique contemporaine. Au cours des trente dernières années, les vapeurs diluées d’atomes neutres en suspension dans le vide et contrôlées par un laser sont devenues une plate-forme polyvalente et formidable pour l’étude de tels systèmes. L’intérêt principal réside dans la capacité d’ajuster arbitrairement la force de l’interaction atomique au moyen de résonances de Feshbach induites magnétiquement, ainsi que la possibilité de créer une large gamme de potentiels via des champs optiques précisément adaptés. Cette thèse présente les résultats récents de l’expérience FerMix, consacrée à l’étude des systèmes quantiques à plusieurs corps fermioniques à des températures ultra-basses utilisant les atomes alcalins 40K et 6Li. Les principaux résultats présentés dans ce texte sont doubles. Premièrement, nous rapportons la caractérisation expérimentale d’une nouvelle résonance de Feshbach (s,d)-wave du 40K, dont les résultats sont comparés aux prédictions théoriques correspondantes. En particulier, le spectre du taux de perte inélastique est déterminé pour différentes températures et profondeurs de piège, ce qui nous permet d’identifier les pertes en tant que processus à deux corps. De plus, il est confirmé que le canal d’entrée dominant est de type s-wave. À l’aide de modèles d’équation de taux, nous analysons le réchauffement observé de l’ensemble atomique et trouvons que le comportement est cohérent avec l’état lié prévu L = 2 présent dans le canal de sortie. Enfin, nous étudions expérimentalement la dynamique des populations de spin induite par les collisions inélastiques renforcées par résonance dans l’onde d, en observant un bon accord avec nos modèles numériques. En second lieu, nous résumons nos progrès dans l’étude des croisements dimensionnels entre le liquide de Tomonaga-Luttinger en 1D et le liquide de Landau-Fermi en 3D en utilisant les gaz de Fermi de 40K confinés dans un réseau optique à grand pas. Cela inclut à la fois les considérations de conception fondamentales et l’installation du matériel expérimental requis. / The exploration of strongly correlated quantum many-body systems represents one of the most challenging fields of research of contemporary physics. Over the past thirty years, dilute vapors of neutral atoms suspended in vacuum and controlled with laser light have become a versatile and powerful platform for the study of such systems. At the very heart lies the ability to arbitrarily tune the interaction strength by means of magnetically induced Feshbach resonances as well as the possibility to create a wide range of potential landscapes via precisely tailored optical fields. This thesis reports on the recent results of the FerMix experiment, which is dedicated to the study of fermionic quantum many-body-systems at ultralow temperatures using the Alkali atoms 40K and 6Li. The main results presented in this text are twofold. First, we report on the experimental characterization of a novel (s,d)-wave Feshbach resonance in 6Li, the results of which are compared to the corresponding theoretical predictions. In particular, the spectrum of the inelastic loss rate is determined for different temperatures and trap depths, which enables us to identify the losses as two-body processes. Moreover, the dominant entrance channel is confirmed to be s-wave in nature. Using rate equation models we analyze the observed heating of the atomic ensemble and find the behavior to be consistent with the predicted L = 2 bound state present in the exit channel. Finally, we investigate experimentally the dynamics of the spin populations driven by resonantly enhanced inelastic collisions in dwave, observing good agreement with our numerical models. Second, we summarize our progress towards the study of dimensional crossovers between the Tomonaga-Luttinger liquid in 1D and the Landau-Fermi liquid in 3D using Fermi gases of 40K confined in a large spacing optical lattice. This includes both the fundamental design considerations as well as the implementation of the required experimental hardware.

Diese Arbeit beschäftigt sich mit theoretischen Untersuchung von Oberflächenzuständen in ferromagnetischen Halbleitern. Einleitend wird ein analytisches "tight-binding"-Modell zur Beschreibung der Oberflächenzustände. Es liefert Aussagen zur Existenz von Oberflächenzuständen, zu deren spektralen Gewicht und Position bezüglich der Energie. Das Kondogitter-Modell wird verwendet, um Korrelations- und Temperatureffekte sowie Oberflächenzustände zu beschreiben. Dies erfolgt zunächst für sc-(100)-Modellfilme im Rahmen des sf-Modells. Die temperaturabhängigen der Oberflächenzustände zeigen abhängig vom Ort in der Brillouinzone sowie den "hopping"-Parametern in der Oberfläche sowohl Stoner-artiges als auch "spin-mixing"-Verhalten. Mit wachsender Temperatur werden Lebensdauereffekte in den Spektren sichtbar. Das Kondogitter-Modell (KLM) wird auf die Mehrbandsituation zum df-Modell verallgemeinert, um eine Beschreibung der Prototypen für magnetische Halbleiter EuS und EuO zu erreichen. Durch die Kombination von LDA-Bandstrukturrechnungen mit Vielteilchenrechnungen zum Multiband-KLM ist es gelungen, die ausgeprägte Temperaturabhängigkeit des unbesetzten 5d-Leitungsbandes und das Verhalten der Oberflächenzustände in EuS- und EuO-Filmen realistisch zu beschreiben. Der exakte Grenzfall des Kondogitter-Modells, das magnetischen Polaron (T=0), ermöglicht Kombination von ab-initio-Bandstrukturrechnungen und der Vielteilchentheorie ohne das Auftreten von Doppelzählungen relevanter Wechselwirkungen. Sowohl in EuO als auch in EuS temperaturabhängige Oberflächenzustände beobachtet werden, die im Fall von EuS jedoch schwerer nachzuweisen sind, da sie im Energiebereich des Volumenbandes auftreten. Die für EuS und EuO berechneten Rotverschiebungen sowie die dickenabhängige Magnetisierung von EuS stimmen hervorragend mit experimentellen Befunden überein. Eine Vielzahl von Korrelationseffekten ist mit wachsender Temperatur in den Spektren der unbesetzten Europium-5d-Bänder zu beobachten. / This work is dedicated to the theoretical investigation of surface states in ferromagnetic semiconductors. After the introduction a exact solvable analytical model is presented. It figures out for given conditions if surface states exist, which spectral weight they have, and at which position in energy they can be found. Thereafter the Kondo-Lattice-Model (KLM) is used to describe correlation and temperature effects. The description focuses initially to the sc-(100) model films in the sf-model. The resulting temperature dependent surface states both Stoner behavior and "spin-mixing" behavior dependent on the chosen hopping parameters and the position in the two dimensional Brillouin zone. With increasing temperature (up to Tc) lifetime effects arise in the spectra. In conclusion the KLM is extended to the multi-band situation (df-Model) in order to provide a description of the prototypes of magnetic semiconductor EuS and EuO. We succeed in describing the distinct temperature dependence of the unoccupied 5d-conduction band and the behavior of the surface states in EuS and EuO films realistically by a combination of a LDA band-structure calculation and the manybody theory. The exact limiting case of the KLM (T=0) -the magnetic polaron- allows a combination of a ab-initio band-structure calculation and manybody theory without double counting of any relevant interaction. The presented theory provides numerous results. In EuO and EuS can be found temperature dependent surface states. In case of EuS it''s detection is much more complicated thus the surface state energies are located inside the energy range of the bulk band. The famous redshift in EuS and EuO and the thickness dependent film magnetization of EuS agree very well with the experimental results. A lot of correlation effects are present in the calculated unoccupied Europium 5d bands. With increasing temperature these effects become stronger.