The inter-dimensional system of a three-dimensional bulk and a two-dimensional interface is analyzed for bosons with quasirelativistic dispersion relation. Using the energy-dependent Green's function, the inter-dimensional quasirelativistic density of states inside the interface is solved analytically. We find that in the low-energy limit the inter-dimensional quasirelativistic density of states shows three-dimensional behaviour, while in the high-energy limit the density of states shows two-dimensional behaviour. This work is the first step in a process that will lead to a fully-analytical model for fermions in topological materials.

[pt] O objetivo deste trabalho é o estudo das condições para a estabilidade de sistemas lineares discretos de dimensão infinita invariantes no tempo, evoluindo em um espaço de Hilbert. Apresentaremos uma vasta coleção de resultados sobre estabilidade assintótica uniforme, incluindo uma condição espectral equivalente. Em relação à estabilidade assintótica fraca, analisaremos tanto a dificuldade de se estabelecer uma condição necessária e suficiente sobre o espectro do operador, como também sua relação com similaridade a contração. Por último, apresentaremos alguns resultados disponíveis sobre estabilidade assintótica forte para algumas classes específicas de operadores. / [en] The purpose of this work is to analyse stability conditions for infinity-dimensional linear discrete systems operating in a Hilbert space. Whe shall present a wide collections of results on uniform asymptotic stability, incluiding an equivalent spectral condition. Concerning the weak asymptotic stability, we shall analyse the dificulty associated to the problem of attempting to establish a necessary and sufficient condition involving the spectral of the system operator. The relation between weak asymptotic stability and similarity to a contraction will be analysed as well. Finally, we shall present some of the available results concerning strong asymptotic stability for particular classes of operators.

Ferromagnetismo itinerante permanece um problema elusivo em Física. O fenômeno resulta da competição entre interação eletrônica e efeitos de muitos corpos e não pode ser tratado perturbativamente. Particularmente em uma dimensão, teoremas proíbem fases ferromagnéticas em T = 0 para modelos de rede com hopping de primeiros vizinhos. Nos últimos vinte anos, entretanto, apareceram modelos na literatura que estendem o hopping para além de primeiros vizinhos e para os quais ordem ferromagnética foi rigorosamente estabelecida. Praticamente todas as demonstrações da existência de ferromagnetos unidimensionais são feitas em fase isolante (com exceção de casos patológicos, como repulsão infinita). Isto nos levou a investigar o acoplamento entre os setores de spin e carga no regime fortemente interagente quando se dopa o sistema, o que introduz pontos de Fermi pF e -pF. Encontramos, com teoria de perturbação, singularidades logarítmicas na autoenergia do mágnon quando seu momentum é pF ou -pF. Derivamos uma teoria de campo efetiva para o espalhamento em torno desses pontos entre os mágnons e férmions sem spin (que representam o setor de carga). O modelo efetivo é similar ao modelo Kondo, que consiste de uma impureza magnética localizada acoplada localmente com um mar fermiônico por uma interação de troca entre spins. Em nosso modelo, há, na realidade, um pseudospin que indica se o momentum de uma partícula é próximo de pF ou de -pF e o mágnon se comporta como uma impureza móvel. A mobilidade da impureza leva a uma relação de dispersão para os férmions dependente do pseudospin da impureza. / Itinerant ferromagnetism remains an elusive problem in Physics. The phenomenon arises from a competition between electronic interaction and many-body effects and cannot be treated perturbatively. Particularly in 1D, there are rigorous proofs that forbid ferromagnetic phase for lattice models with nearest-neighbours hopping only. In the last twenty years, however, models with hopping beyond nearest-neighbours were proposed in the literature and for which ferromagnetic phase was rigorously established. Virtually every proof of the existence of one-dimensional ferromagnets is done in an insulator phase (disregarding some pathological cases, such as infinite electronic repulsion). That motivated us to investigate the coupling between spin and charge sectors in the strongly interacting regime when we dope the system, introducing two Fermi points, pF and -pF. We found out, through perturbation theory, logarithmic singularities in the magnon selfenergy when its momentum is pF or -pF. To understand them, we derived an effective field theory for the scattering between magnons and spinless fermions (which represent the charge sector) close to these points. The effective model resembles the Kondo model, which describes a magnetic impurity locally coupled to a fermionic sea through spin exchange interaction. In our model, there is actually a pseudospin that indicates if a particle momentum is closest to pF or -pF and the magnon behaves as a mobile impurity. The impurity mobility leads to a fermionic dispersion relation that depends on the impurity pseudospin.

京都大学 / 0048 / 新制・課程博士 / 博士(人間・環境学) / 甲第22546号 / 人博第949号 / 新制||人||226(附属図書館) / 2019||人博||949(吉田南総合図書館) / 京都大学大学院人間・環境学研究科相関環境学専攻 / (主査)准教授 木下 俊哉, 教授 吉田 鉄平, 教授 森成 隆夫 / 学位規則第4条第1項該当 / Doctor of Human and Environmental Studies / Kyoto University / DGAM

Thesis advisor: Michael J. Naughton / In this dissertation I present studies of plasmonic interactions in different coupling regimes, from zero to strong coupling and approaching ultrastrong coupling. Different physics are manifest in each regime, with different possible applications. The first project uses finite element electromagnetic simulations to model plasmonic waveguides that couple near field light into the far-field for sub-diffraction limited microscopy. Wavelength/32 resolution is shown by minimizing coupling between adjacent waveguiding nanowires, with minimal attenuation over a few microns. The next two projects, by contrast, seek to maximize coupling between plasmons and excitons into the strong coupling regime where the optoelectronic properties are modified and quantum coherent phenomena may be observed. Strong exciton–plasmon coupling in MoS2 is shown experimentally at room temperature and found to be a general phenomenon in other semiconducting transition metal dichalcogenides using transfer matrix modeling. A semiclassical oscillator model is fit to the experimental data to discover coherent hybridization between the ground and first excited states of MoS2. Enhanced coupling is found at the third excitonic transition, approaching the ultrastrong coupling regime where exotic properties are predicted to emerge, such as ground state virtual photons. Our strong coupling studies motivate further studies of the TMDCs as a platform for coherent quantum physics with possible applications in quantum computing and cryptography. / Thesis (PhD) — Boston College, 2019. / Submitted to: Boston College. Graduate School of Arts and Sciences. / Discipline: Physics.

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The high-dimensional property of delay differential equations makes them useful for various purposes. The applications of systems modelled with delay differential equations demand different degrees of complexity. One solution to tune this property is to make the dynamics of the current state dependent on more delayed states. How the system responds to more delayed states depends on the system under study, as both decreases and increases in the complexity were observed in different nonlinear systems. However, it is also known that when there is an infinite number of delays that follow a continuous distribution, simpler dynamics usually expected due to the averaging over previous states that the delay kernel provides. The present thesis investigates the role of multiple delays in nonlinear time delay systems, as well as methods for evaluating their complexity. Through the use of pseudospectral differentiation, we first compute the Lyapunov exponents of such multi-delay systems. In systems with a large number of delays, chaos is found to be less likely to occur. However, in systems with oscillatory feedback functions, the entropy can increase just by adding a few delays. Our study also demonstrates that the transition to simpler dynamics in nonlinear delay systems can be either monotonous or abrupt. This is particularly true in first-order nonlinear systems, where increasing the width of the distribution of delays results in complexity collapse, even in the presence of a few discrete delays. The roots of the characteristic equation around a fixed point can be used to approximate the degree of complexity of the dynamics of such time-delay systems, as they can be linked to other dynamical invariants such as the Kolmogorov-Sinai entropy. The phenomenon of complexity collapse uncovered in our work was further studied in an 80/20 ratio excitatory-inhibitory neural network. We found that the smaller the time delay, the higher the likelihood of chaotic dynamics, and this also promotes asynchronous spiking activity. But for larger values of the delay, the neurons show synchronized oscillatory spiking activity. A global inhibition at a longer delay results in a novel dynamical pattern of randomly occurring epochs of synchrony within the chaotic dynamics. The final part of the thesis examines the behavior of time delay reservoir computing when there are multiple time delays. It is shown that the choice of spacing between time delays is crucial, and depends on the task at hand. The system was studied for a prediction task with one chaotic input as well as for a mixture of two chaotic inputs. It was found that, similar to the single delay case, there is a resonance when the difference between delays is equal to the clock cycle. Together, our research provides valuable insights into the dynamics and complexity of nonlinear multi-delay systems and the importance of the spacing between delays.

Complex and networked dynamical systems characterize the time evolution of most of the natural and human-made world. The dimension of their state space, i.e., the number of (active) variables in such systems, arguably constitutes their most fundamental property yet is hard to access in general. Recent work [Haehne et al., Phys. Rev. Lett. 122, 158301 (2019)] introduced a method of inferring the state space dimension of a multi-dimensional networked system from repeatedly measuring time series of only some fraction of observed variables, while all other variables are hidden. Here, we show how time series observations of one single variable are mathematically sufficient for dimension inference. We reveal how successful inference in practice depends on numerical constraints of data evaluation and on experimental choices, in particular the sampling intervals and the total duration of observations. We illustrate robust inference for systems of up to N = 10 to N = 100 variables by evaluating time series observations of a single variable. We discuss how the faithfulness of the inference depends on the quality and quantity of collected data and formulate some general rules of thumb on how to approach the measurement of a given system.

Dans cette thèse nous avons étudié des indicateurs visant à caractériser les propriétés métalliques ou isolantes de systèmes de basse dimensionnalité à partir de calculs théoriques basés sur la fonction d'onde. Ces systèmes sont intéressants car ils permettent une compréhension en profondeur des phénomènes physiques qui peuvent ensuite être extrapolés à des systèmes plus étendus. Afin de réaliser cette étude nous avons utilisé un nouvel outil basé sur la théorie de la conductivité de Kohn : le tenseur de délocalisation total ou total position spread-tensor (TPS). Ce tenseur est défini comme le second cumulant de l'opérateur position : ? = <?|X2|?> - <?|X|?>2. Divisé par le numéro des électrons, il diverge quand la fonction d'onde est fortement délocalisée (forte fluctuation de la position des électrons) et converge vers une valeur finie dans le cas contraire. Ainsi, la conductivité est relié à la délocalisation de la fonction d'onde. Dans ce travail, deux définitions du TPS ont été abordées : une quantité sommée sur le spin (spin-summed TPS, SS-TPS) d'une part, et une décomposition selon le spin (spin-partitioned TPS, SSP-TSP) d'autre part. Cette dernière s'est avérée être un outil très efficace pour l'étude de systèmes fortement corrélés. Au cours de la thèse, nous avons commencé par étudier plusieurs systèmes diatomiques présentant des liaisons de natures différentes à l'aide de calculs d'interaction de configurations totale (FCI). Le TPS présente alors un maximum dans une zone précédant la rupture de liaison avant de converger asymptotiquement vers les valeurs atomiques, comme la consistance de taille du tenseur le laissait présager. Dans le cas de systèmes pour lesquels l'état électronique présente un croisement évité, le TPS diverge, mettant ainsi en évidence la forte délocalisation de la fonction d'onde. Le SS-TPS est donc un indicateur de choix pour suivre la nature de la liaison chimique. Nous avons ensuite considéré des systèmes à valence mixte de type II pour lesquels l'état fondamental présente un double-puits de potentiel avec un croisement évité avec le premier état excité. Il est donc nécessaire ici d'utiliser un traitement multi-configurationnel. Deux systèmes modèles ont ainsi été étudiés : i) deux di- mères H2 en interaction faible au niveau FCI et ii) un composé du type spiro au niveau CAS-SCF (à l'aide d'un code que nous avons implémenté dans Molpro). Dans les deux cas, le TPS présentait un maximum très marqué dans la région du croisement évité, signature d'une forte mobilité électronique. Nous nous sommes également intéressés à trois types de chaines d'atomes d'hydrogène : i atomes équidistants ii) chaines dimérisées à longueur de liaison H2 fixée et iii) chaines dimérisées. Tant le SS-TPS que le SP-TPS montrent des comportements différents selon le type de chaine considérée. Les premières ont un caractère métallique et une délocalisation de spin prononcée dans le régime fortement corrélé. Les secondes sont de nature isolante avec une délocalisation limitée. Les chaines dimérisées, quant à elle, dissocient très rapidement vers un état isolant mais avec une forte délocalisation de spin. Ces chaines demi-remplies ont aussi été traitées à l'aide d'hamiltonien de Hubbard et de Heisenberg. Nous avons ainsi pu rationaliser le comportement des SS-TPS et SP-TPS en variant le rapport de l'intégrale de saut et de la répulsion électron- électron (-t/U) entre sites adjacents. Le caractère ferromagnétique/anti-ferromagnétique a également pu être suivi en modifiant la valeur de la constante de couplage J dans le cas fortement corrélé. Finalement, ces indicateurs ont été mis en oeuvre pour des polyacenes cycliques. Dans ce cas, le TPS a permis de comprendre la nature des fonctions d'onde de l'état fondamental obtenues au niveau CAS-SCF et NEVPT2. / I carried out a theoretical study to characterize metallic and insulating properties of low-dimensional systems using wave function methods. Low-dimensional systems are particularly important because they allow an understanding that can be extrapolated to higher dimensional systems. We have employed a new tool based on the theory of conductivity of Kohn that we have named: total position-spread tensor (TPS). The TPS is defined as the second moment cumulant of the total position operator: ? = <?|X2|?> - <?|X|?>2 . The tensor divided by the number of electrons diverges when the wave function is delocalized (high fluctuation of electrons' positions), and it takes finite values for localized ones. In this way, the electrical conductivity is related to the proper delocalization of the wave function. In addition, the tensor can be divided in spin-summed (SS-TPS) and spin-partitioned tensors (SP-TPS). The latter one becomes a powerful tool to the study of strongly correlated systems. In this dissertation, we started to investigate at full configuration interaction (FCI) level diatomic molecules showing different types of bond. The TPS presented a marked maximum before the bond was broken and in the asymptotic limit one recovers the TPS values of isolated atoms (size consistency). For the case of diatomic systems showing avoided-crossing electronic states, the TPS diverges evidencing the high delocalization of the wave function. Therefore, the SS-TPS is capable of monitoring and characterizing molecular wave functions. We considered mixed-valence systems that are often distinguished by a double-well potential energy surface presenting an avoided-crossing. Thus, such a configuration possesses a strongly multireference nature involving at least two states of the same symmetry. Two different systems were investigated: i) two weakly interacting hydrogen dimers that were investigated at Full CI level, and ii) a spiro like molecule where the TPS tensor was evaluated in a CAS-SCF state-averaged wave function using our implementation of the SS- TPS formalism in MOLPRO. We found that the tensor's component in the direction of the electron transfer (ET) shows a marked maximum in the avoided-crossing region, evidencing the presence of a high electron mobility. The formalisms of the SS- and SP-TPS was applied to one dimensional systems composed by three types of half-filled hydrogen chains: i) equally-spaced chains, ii) fixed-bond dimerized chains, and iii) homothetic dimerized chains. Both the SS- and SP-TPS showed different signatures associated to the three types of systems. Equally-spaced chains have metallic wave functions and a high spin delocalization in the strongly correlated regime. In contrast, fixed-bond dimerized chains have an insulating character and a restricted spin delocalization. Finally, homothetic dimerized chains dissociate very quickly which renders them in the insulating state but with a high spin delocalization. We also studied half-filled chains by using the Hubbard and the Heisenberg Hamiltonians. On the one hand, we were able to depict the response of the SS- and SP-TPS by varying the ratio between the hopping and electron-electron repulsion (-t/U parameter) of topological connected sites. On the other hand, the ferromagnetic and anti-ferromagnetic character of the wave functions were evaluated by varying the coupling constant (J) in the strongly correlated systems. A theoretical study of closed polyacenes (PAH) structures was performed at CAS-SCF and NEVPT2 level. Our methodology for choosing the active space using the Hückel Hamiltonian was able to characterize the ground state of the systems that indeed fulfilled the Ovchinnikov rule. Finally, we applied the SS-TPS to understand the nature of the wave functions of these PAHs.

Ce travail de thèse est consacré à l'étude sur une large gamme de températures d'un système unidimensionnel modèle, les nano-peapods de carbone. Ces composés sont constitués de fullerènes (C60, dans notre cas) insérés dans des nanotubes de carbone monofeuillets. Les diamètres des fullerènes et des tubes étant concordant, les fullerènes s'arrangent selon un réseau 1D.Dans le premier chapitre de ce manuscrit, nous décrivons la synthèse des peapods de carbone. Dans les deux chapitres suivants, nous décrivons les différents modèles et méthodes expérimentales qui nous permettent de déterminer l'évolution de la structure moyenne des chaînes de fullerènes, ainsi que la dynamique de rotation et de translation des molécules.Dans les trois derniers chapitres, nous décrivons l'évolution de la structure et de la dynamique des chaînes sur trois gammes de températures, que nous appelons hautes (500-1100 K), basses (0-200 K), et intermédiaires (200-500 K). Les résultats expérimentaux concernant deux types d'échantillons, les peapods monomères et les peapods polymères (dans lesquels les degrés de liberté de rotation sont restreints) sont confrontés à des modèles analytiques. Nous mettons en évidence trois comportements différents des chaînes dans ces trois gammes de température. / This work is dedicated to the study of the structural and dynamical behavior of a model one-dimensional system over a wide temperature range: carbon nano-peapods. This compound is constituted of fullerenes (C60, in our case) inserted inside single-walled carbon nanotubes. The perfect match between the inner diameter of the tubes and the diameter of the fullerenes results in a chain organization of the C60 molecules.The synthesis of these peapods is described in the first part of this manuscript. The two next chapters are aimed to the description of the different experimental and simulation methods that are used to monitor the structural and dynamical behavior of the C60 molecules.In the three last chapters, we describe the behaviour of the C60 molecules over three temperature ranges, labeled high (500-1100 K), low (0-200 K), and intermediary (200-500 K) ranges. By comparing experimental results to analytical models for both monomer and polymer peapods (the rotational degree of freedom being hindered in the latter), we highlight three different behaviors of the molecules in these three ranges.