The scope of this thesis is analyzing and characterizing certain gapless states in tight-binding non-Hermitian systems. We start by providing a pedagogical introduction to tight-binding theory, topological phases of matter, Wannier functions as real-space duals of Bloch functions and their properties, non-Hermitian systems and associated differences from standard Hermitian systems. Subsequently we show the possibility of extending pre-existing concepts of Hermitian quantum mechanics to non-Hermitian settings without losing predicting power over some peculiar observables. We conclude by providing numerical evidence for existence of certain topological states in finite one-dimensional and two-dimensional systems, also testing their robustness against symmetry-breaking and disorder.

The field of condensed matter computational physics has seen an explosion of applicability over the last 50+ years. Since the very first calculations with ENIAC and MANIAC the field has continued to pushed the boundaries of what is possible; from the first large-scale molecular dynamics simulation, to the implementation of Density Functional Theory and large scale Car-Parrinello molecular dynamics, to million-core turbulence calculations by Standford. These milestones represent not only technological advances but theoretical breakthroughs and algorithmic improvements as well. The work in this thesis was completed in the hopes of furthering such advancement, even by a small fraction. Here we will focus mainly on the calculation of electronic and structural properties of solids and liquids, where we shall implement a wide range of novel approaches that are both computational efficient and physically enlightening. To this end we routinely will work with maximally localized Wannier functions (MLWFs) which have recently seen a revival in mainstream scientific literature. MLWFs present us with interesting opportunity to calculate a localized orbital within the planewave formalism of atomistic simulations. Such a localization will prove to be invaluable in the construction of layer-based superlattice models, linear scaling hybrid functional schemes and model quasiparticle calculations. In the first application of MLWF we will look at modeling functional piezoelectricity in superlattices. Based on the locality principle of insulating superlattices, we apply the method of Wu et al to the piezoelectric strains of individual layers under iifixed displacement field. For a superlattice of arbitrary stacking sequence an accurate model is acquired for predicting piezoelectricity. By applying the model in the superlattices where ferroelectric and antiferrodistortive modes are in competition, functional piezoelectricity can be achieved. A strong nonlinear effect is observed and can be further engineered in the PbTiO 3 /SrTiO 3 superlattice and an interface enhancement of piezoelectricity is found in the BaTiO 3 /CaTiO 3 superlattice. The second project will look at The ionization potential distributions of hydrated hydroxide and hydronium which are computed within a many-body approach for electron excitations using configurations generated by ab initio molecular dynamics. The experimental features are well reproduced and found to be closely related to the molecular excitations. In the stable configurations, the ionization potential is mainly perturbed by solvent water molecules within the first solvation shell. On the other hand, electron excitation is delocalized on both proton receiving and donating complex during proton transfer, which shifts the excitation energies and broadens the spectra for both hydrated ions. The third project represents a work in progress, where we also make use of the previous electron excitation theory applied to ab initio x-ray emission spectroscopy. In this case we make use of a novel method to include the ultrafast core-hole electron dynamics present in such situations. At present we have shown only strong qualitative agreement with experiment. / Physics

The importance and promise that topological materials hold has been recently underscored by the award of the Nobel Prize in Physics in 2016 ``for theoretical discoveries of topological phase transitions and topological phases of matter." This dissertation explores the novel qualities and useful topologically protected surface states of topological insulators and semimetals. Topological materials have protected qualities which are not removed by weak perturbations. The manifestations of these qualities in topological insulators are spin-momentum-locked surface states, and in Weyl and Dirac semimetals they are unconventional open surface states (Fermi arcs) with anomalous electrical transport properties. There is great promise in utilizing the topologically protected surface states in electronics of the future, including spintronics, quantum computers, and highly sensitive devices. Physicists and chemists are also interested in the fundamental physics and exotic fermions exhibited in topological materials and in heterostructures including them. Chapter 1 provides an introduction to the concepts and methods of topological band theory. Chapter 2 investigates the spin and spin-orbital texture and electronic structures of the surface states at side surfaces of a topological insulator, Bi2Se3, by using slab models within density functional theory. Two representative, experimentally achieved surfaces are examined, and it is shown that careful consideration of the crystal symmetry is necessary to understand the physics of the surface state Dirac cones at these surfaces. This advances the existing literature by properly taking into account surface relaxation and symmetry beyond what is contained in effective bulk model Hamiltonians. Chapter 3 examines the Fermi arcs of a topological Dirac semimetal (DSM) in the presence of asymmetric charge transfer, of the kind which would be present in heterostructures. Asymmetric charge transfer allows one to accurately identify the projections of Dirac nodes despite the existence of a band gap and to engineer the properties of the Fermi arcs, including spin texture. Chapter 4 investigates the effect of an external magnetic field applied to a DSM. The breaking of time reversal symmetry splits the Dirac nodes into topologically charged Weyl nodes which exhibit Fermi arcs as well as conventionally-closed surface states as one varies the chemical potential. / Ph. D.

The electronic structure of the zero-gap two-dimensional graphene has a charge neutrality point exactly at the Fermi level that limits the practical application of this material. There are several ways to modify the Fermi-level-region of graphene, e.g. adsorption of graphene on different substrates or different molecules on its surface. In all cases the so-called dispersion or van der Waals interactions can play a crucial role in the mechanism, which describes the modification of electronic structure of graphene. The adsorption of water on graphene is not very accurately reproduced in the standard density functional theory (DFT) calculations and highly-accurate quantum-chemical treatments are required. A possibility to apply wavefunction-based methods to extended systems is the use of local correlation schemes. The adsorption energies obtained in the present work by means of CCSD(T) are much higher in magnitude than the values calculated with standard DFT functional although they agree that physisorption is observed. The obtained results are compared with the values available in the literature for binding of water on the graphene-like substrates. / Dieser Beitrag ist mit Zustimmung des Rechteinhabers aufgrund einer (DFG-geförderten) Allianz- bzw. Nationallizenz frei zugänglich.

Le travail présenté dans ce manuscrit porte sur deux sujets distincts. Le premier concerne la réponse d'un diélectrique cristallin à un champ électrique uniforme ; il s'ancre sur la théorie moderne de la polarisation développée par King-Smith, Vanderbilt et Resta. En nous restreignant d'abord au cas unidimensionnel, nous décrivons de manière perturbative à faible champ électrique le spectre de Wannier-Stark d'un modèle de bande. Nous utilisons ensuite ce développement dans une approche thermodynamique que nous modifions pour palier aux problèmes posés par le caractère non-borné du spectre de Wannier-Stark : nous introduisons en particulier un potentiel chimique local assurant la neutralité électrique locale au sein du cristal. Cette approche permet d'accéder à la polarisation et à la susceptibilité électrique des cristaux diélectriques. Finalement, nous étendons le travail effectué au cas bidimensionnel où de nouvelles caractéristiques associé aux isolants topologiques apparaissent.Le deuxième sujet porte sur la synchronisation de spin dans les gaz d'atomes froids. Nous étudions la compétition entre le mécanisme d'écho de spin et le phénomène d'auto-synchronisation lié à l'effet de rotation des spins identiques (emph{ISRE}). La méthode de l'écho de spin permet de compenser certains déphasage apparaissant dans une gaz d'atomes ultra-froid piégé, et accroît ainsi le temps de cohérence de l'ensemble. L'emph{ISRE} apparaît dans les gaz denses via les collisions entre atomes et conduit également à un accroissement du temps de cohérence. Nous montrons que ces deux mécanismes ne sont pas systématiquement compatibles. En particulier, leur compatibilité est lié à la relation entre les échelles de temps propres à chacun des phénomènes. / The work exposed in this manuscript covers two distinct topics. The first is about the response of crystalline dielectrics to an external static electric field; it is based on King-Smith, Vanderbilt and Resta modern theory of polarisation. Restricting ourselves to the 1D case, we first describe the Wannier-Stark ladder of a band model with a low-field perturbative approach. We then use this development to derive the thermodynamical response of the band model. We have to modify the usual thermodynamics to account for the unboundedness of the Wannier-Stark spectrum, through the introduction of a local chemical potentiel which ensures local electric neutrality in the crystal. In a last step, we extend our approch to the 2D cas, where new characteristics related to the topic of topological insulators appear.The second topic tackles synchronization and spin-echo in cold atom gases. We study the competition between the spin-echo mechanism and the self-synchronization mechanism which emerges from the identical spin rotation effet (emph{ISRE}). The spin-echo thechnique was built to compensate for some the of dephasing that appears in trapped ultra-cold gases, leading to an increased coherence time for the ensemble. The emph{ISRE} appears in dense atomic clouds where collisions also lead to an increased coherence time. We show that these two mechanism are not always compatible, in particular, their compatibility is based on the relation between the time scales associated to both phenomena.

The electronic structure of the zero-gap two-dimensional graphene has a charge neutrality point exactly at the Fermi level that limits the practical application of this material. There are several ways to modify the Fermi-level-region of graphene, e.g. adsorption of graphene on different substrates or different molecules on its surface. In all cases the so-called dispersion or van der Waals interactions can play a crucial role in the mechanism, which describes the modification of electronic structure of graphene. The adsorption of water on graphene is not very accurately reproduced in the standard density functional theory (DFT) calculations and highly-accurate quantum-chemical treatments are required. A possibility to apply wavefunction-based methods to extended systems is the use of local correlation schemes. The adsorption energies obtained in the present work by means of CCSD(T) are much higher in magnitude than the values calculated with standard DFT functional although they agree that physisorption is observed. The obtained results are compared with the values available in the literature for binding of water on the graphene-like substrates. / Dieser Beitrag ist mit Zustimmung des Rechteinhabers aufgrund einer (DFG-geförderten) Allianz- bzw. Nationallizenz frei zugänglich.

Cette thèse présente une étude théorique des oscillations de courant et l'émission de lumière THz dans les super-réseaux de semi-conducteur polarisés soumis à une excitation optique pulsée. Ce travail a été fait en étroite collaboration avec des expérimentateurs. Ceci a permis de faire des comparaisons entre la théorie et l'expérience. Dans la dernière partie de cette thèse, nous poursuivons notre étude en incluant les effets sur les oscillations de courant d'un champ magnétique parallèle au champ électrique de polarisation. Nous modélisons par une approche quantique la dynamique des porteurs de charge dans les super-réseaux lors d'une excitation optique femtosecondes au moyen des états de Wannier-Stark et excitoniques. Tous les paramètres mis en jeu dans le phénomène des oscillations de courant sont discutés. De même, nous donnons les éléments nécessaires à l'optimisation de l'émission THz. L'excellent accord entre les calculs et les résultats expérimentaux donne crédit à notre modélisation. Finalement, nous incluons les effets sur l'amplitude du courant et de l'émission suite à l'application d'un champ magnétique intense appliqué parallèlement au champ de polarisation. Nous prédisons d'importantes modifications du spectre des oscillateurs, associées à la quantification additionnelle du quasi-continuum existant dans le plan perpendiculaire au champ électrique en l'absence du champ magnétique.

Orientador: Alexys Bruno Alfonso / Banca: Ernesto Reyes Gómez / Banca: Fabio de Jesus Ribeiro / O programa de Pós graduação em Ciência e Tecnologia de Materiais, PosMat, tem carater institucional e integra as atividades de pesquisa em materiais de diversos campi da UNESP / Resumo: Os cristais fotônicos são novos materiais em que a permissividade dielétrica e a permeabilidade magnética apresentam a periodicidade de uma rede de Bravais. No caso dos sólidos cristalinos, é a energia potencial de cada elétron que apresenta esse tipo de periodicidade. Por conta das semelhanças que existem entre as equações que descrevem os sintomas eletrônicos e fotônicos, muitos dos conceitos e métodos da teoria de estados eletrônicos vêm sendo aplicados na investigação de cristais fotônicos. Dentre esses conceitos, as funções de Wannier apresentam vantagens para o tratamento de estados eletrônicos e modos fotônicos localizados. Este trabalho aborda o caso de cristais fotônicos unidimensionais com simetria de inversão. Primeiramente são calculados e analisadas funções de Wannier bem localizadas. Em seguida são investigados os modos eletromagnéticos localizados produzidos por defeitos em cristais fotônicos, utilizando o método de matriz de transferência e o método da combinação linear das funções de Wannier. Finalmente, é feita a comparação dos resultados obtidos mediante esses métodos e são discutidas as vantagens do uso de funções de Wannier / Abstract: Photonic crystal are new materials where both the dielectric permittivity and the magnetic permeability present the periodicity of a Bravais lattice. In the case of electronic states, this property is shown by the mono-eletronic potential energy. Due to the similarities between the eletronic and photonic problems, many concepts and methods of theory of electronic states are being applied to investigate photonic crystals. Among such concepts, the Wannier functions are advantageous to deal with either localized photonic modes. This work addresses the case of one-dimensional photonic crystals with inversion symmetry. First, well-localized Wannier functions are calculated and analyzed. Then, the localized modes produced by defects in photonic crystals are investigated by using either transfer matrices or linear combination Wannier functions. Finally, the results obtained by the two methods are compared and the advantages of using Wannier functions are discussed / Mestre

Ce travail de thèse présente la modélisation théorique de l'expérience FORCA-G. L'objectif de cette expérience est la mesure des interactions à courte portée entre des atomes piégés dans un réseau optique et une surface massive à une grande précision. Nous nous sommes intéressés plus particulièrement à l'effet Casimir-Polder induit par la surface sur les atomes. Le but était de fournir la prédiction la plus précise possible des états atomiques. Ceci a consisté à considérer les effets de la température sur l'interaction Casimir-Polder et modéliser la surface de la manière la plus réaliste possible. Afin de résoudre le problème de divergence qu'impliquait un traitement perturbatif de l'interaction atome-surface, nous avons développé une méthode numérique pour un traitement non-perturbatif de l'interaction Casimir-Polder et modélisé l'interaction atome-surface à très courte distance par un potentiel de Lennard-Jones. Chaque effet et incertitude sur les états atomiques ont été évalués afin de déterminer s'ils seraient observables ou un facteur limitant en regard de la précision visée par l'expérience. Enfin nous nous sommes intéressés au cas d'un déséquilibre thermique entre la température du miroir et la température de l'environnement qui pourrait être induit par les lasers en présence ou un laser de chauffage. Nous avons calculé la correction du potentiel Casimir-Polder due au déséquilibre et évalué l'effet sur les niveaux d'énergie atomiques pour déterminer si cet effet pouvait être mesuré. / This thesis presents the theoretical modeling of the experiment FORCA-G. The purpose of this experiment is to measure short-range interactions between trapped atoms in an optical lattice and a massive surface with a high precision. We are focused on Casimir-Polder effect induced by the surface on the atoms. The aim was to give the most possible precise prediction of atomic states. This work took the temperature effects on Casimir-Polder interaction into account, modelled the surface of the experiment. In order to solve the divergence problem due to the perturbative treatment of the atom-surface interaction, we developed a digital method for a non-perturbative treatment of the Casimir-Polder interaction and modelled the short-range atom-surface interaction by a Lennard-Jones potential. Each effect and uncertainties on the atomic states were evaluated so that we know if they could be observable or a limiting factor compared to the experiment precision. Finally we were focused on an out of thermal equilibrium situation between the miroir and environment temperature which may be induced by the lasers. We computed the correction to the Casimir-Polder potential due to this disequilibrium and evaluated the effect on the atomic states.

In 2010 high harmonic generation (HHG) in solids was first observed where high order harmonics of a strong laser field's frequency were observed. HHG in solids is now a rapidly developing field that allows for exciting applications like fully solid state attosecond XUV sources and new ultrafast resolution imaging techniques for quantum dynamics in solids. HHG in solids has been explained by two mechanisms: an interband mechanism, due to polarization associated with separate energy bands, and an intraband mechanism that results from nonlinearities and population changes associated with each individual band. While interband HHG has been seen in wide bandwidth semiconductors, intraband HHG has been observed in narrow bandwidth dielectrics. There has not yet been an explanation of the alternation of mechanisms with material differences. The main goal of this thesis is to attempt to provide a better understanding of the most important mechanisms and where they prevail. Although numerical modelling of HHG requires consideration of multiple energy bands, a two-band model consisting only of a valence band and a single conduction band can explain the most important mechanisms. This model requires a given material's band gap between its valence and conduction bands as well as dipole matrix elements between the bands. In this thesis we follow the Kronig-Penney model to develop a 1D delta-function potential model of solids to obtain these properties required of the two-band model. We implement this in a Wannier quasi-classical (WQC) model of interband HHG in semiconductors that explains the dominant dynamics leading to such through quasi-classical real space electron-hole pair trajectories. Although HHG in solids can be explained to be the result of a resonant process in which an electron-hole pair is generated in the first step, there are also virtual transition processes that lack consideration. These processes do not conserve energy and correspond to transitions to conduction bands resulting from field induced distortions of the ground state. We use methodology introduced by Keldysh for optical field ionization of atoms and solids along with the 1D delta-function potential model to quantify how both resonant and virtual transitions lead to HHG in solids for wide and low bandwidth solids.