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Simulating the nonequilibrium behavior of interacting quantum systems is an important way to understand results of experimental quantum simulators, motivate new materials, and refine new quantum algorithms. However, this is a challenging task due to the exponential difficulty of such systems, which motivates dimensional reduction methods, such as semiclassical limits. This work extends semiclassical phase space methods to spin systems with no clear classical limit with the cluster truncated Wigner approximation (cTWA), and improves on Schrieffer-Wolff low energy effective dynamics with variational adiabatic generators. The cTWA was used to compute nonequilibrium dynamics in spin chains, finding behavior such as signatures of many body localization; rapid thermalization and preservation of fluctuations; effective thermodynamic classical behaviors; and signatures of quantum chaos and butterfly velocities, in 1d spin 1/2 chains. Variational Schrieffer-Wolff methods were used to find efficient non-perturbative dressings for the Hubbard model and find effective quasiparticle dynamics and nonthermal states in quantum chaotic spin chains. These methods are potentially effective tools to separate essential quantum behavior from classical behavior, and can be used to diagnose quantum thermalization behavior in interacting quantum systems.

Thesis: M.S., Massachusetts Institute of Technology, Department of Physics, 1978 / Includes bibiliographical references. / M.S. / M.S. Massachusetts Institute of Technology, Department of Physics

This cumulative dissertation is based on the publications [P1–P6], covering various aspects in theoretical studies of isolated quantum many-body systems. The transport and relaxation dynamics in quantum lattice models are studied with a particular focus on (i) the effect of a mass imbalance between different particles on their relaxation dynamics as well as (ii) the influence of generic perturbations on different reference dynamics. As for (i), the dynamics of two mutually interacting fermionic particle species on a lattice are investigated for different mass ratios between the two species [P4]. Numerical studies of density dynamics show that diffusive transport which is expected for small mass imbalances persists also for moderate imbalances and becomes anomalous for stronger imbalances. On the other hand, while transport is suppressed in the limit of infinite imbalance, i.e., if one particle species is immobile, this effective localization is shown to give way to anomalous diffusion as soon as the heavy particle species gains a finite mobility. Regarding (ii), the effect of perturbations on dynamics is investigated from the perspective of projection-operator techniques [P6]. As a main result, it is demonstrated that simple exponential damping, which is expected in the overwhelming majority of cases, may only occur for the density matrix in the interaction picture. Within this approach, this simple damping carries over to the time dependence of standard correlation functions only in certain cases. In particular, the possibility of nontrivial damping in physically relevant perturbation scenarios is discussed. A considerable portion of this work is concerned with the implementation of powerful numerical and (semi-)analytical tools to overcome the enhanced computational complexity in numerical studies of quantum many-body systems. This includes the concept of dynamical quantum typicality [P2, P3], numerical linked-cluster expansions [P5], and projection-operator techniques, as well as the combined use of available symmetries [P1].

L'objectif de cette thèse est de développer des approximations pour décrire les effets à N-corps dans l'absorption et la photoémission des matériaux avec électrons localisés. Le traitement complet par la mécanique quantique de ce problème difficile repose sur la résolution de l'équation de Schrödinger pour la fonction d'onde à N-corps, ce qui en pratique nécessite des approximations. Pour simplifier, la Théorie de le Fonctionnelle de la Densité (DFT) introduit le système de particules indépendantes de Kohn et Sham. Cependant, il s'avère difficile d'obtenir des propriétés autres que la densité et l'énergie totale. Dans cette thèse, nous travaillons avec des fonctions de Green. Le niveau de complexité de ce cadre, en principe exact, se situe entre la DFT et les méthodes de fonctions d'onde, et de nombreux problèmes restent à résoudre.Quand on décrit l'excitation d'un électron localisé, certaines approximations introduisent une auto-interaction ou auto-écrantage. Ce problème est naturellement évité lorsque l'on utilise une interaction coulombienne généralisée (Chap. 3). De plus, quand l'électron localisé a peu de recouvrement avec les autres électrons, on peut penser que leur interaction est classique. Dans ce cas, l'effet principal à N-corps est la réaction des autres électrons : ils écrantent l'excitation. Dans les approximations habituelles telles que le GW ou la “cumulant expansion”, l'écrantage est traité seulement en réponse linéaire. Cependant, l'excitation d'un électron localisé devrait représenter une forte perturbation. Par conséquent, il se pourrait que les contributions non-linéaires à l'écrantage soient importantes. Comment peut-on vérifier quand cela est vrai ? Et comment peut-on inclure ces effets ? D'autre part, même en réponse linéaire, on pourrait faire mieux que les approximations habituelles, parce que l'écrantage en réponse linéaire est souvent calculé dans l'approximation de la phase aléatoire (RPA). De combien peut-on améliorer les résultats, même en restent en réponse linéaire, si on va au-delà de RPA? Ces points seront abordés dans la thèse.En ce qui concerne l'écrantage, au Chap. 5 on utilise un modèle zéro-dimensionel pour étudier, d'un côté, les effets au-delà de RPA en réponse linéaire, et de l'autre côté, les effets au-delà de la réponse linéaire mais restant en RPA. Fait intéressant, on constate qu'on doit traiter les deux en même temps afin d'obtenir des améliorations significatives. On doit donc trouver des approximations pour aller au-delà de RPA qui sont suffisamment simples pour être utilisées même dans un régime non-linéaire. Dans cette thèse, on développe des approximations basées sur la théorie des perturbations, et on en teste d'autres, déjà existantes, le modèle.L'écrantage est décrit par la fonction diélectrique. Cette fonction permet aussi de calculer les spectres d'absorption. Au Chap.6 on étudie la fonction diélectrique d'un solide modèle à l'aide des fonctions de Wannier localisées. Cela nous permet de mettre en évidence les annulations entre la self-énergie et les effets excitoniques dans le cadre des fonctions de Green et, à partir des résultats, de dériver un potentiel d'échange et corrélation de Kohn-Sham, et un noyau d'échange et corrélation pour la DFT dépendante du temps (TDDFT).Le Chap. 7 aborde la question de comment faire apparaître l'écrantage non-linéaire explicitement dans la formulation ab initio. On propose une réponse possible, en utilisant la localisation de l'électron pour dériver une fonction de Green 'cumulant' au-delà de la réponse linéaire habituelle. On suggère deux niveaux d'approximations pour calculer les expressions en pratique, et on montre quelques résultats préliminaires. Dans les deux cas, la TDDFT est utilisée pour décrire l'écrantage.Etant donné qu'une combinaison de fonctions de Green et de TDDFT semble être une bonne stratégie pour simplifier le problème à N-corps, le Chap. 8 conclut avec quelques idées supplémentaires. / The aim of this thesis is to develop approximations to describe many-body effects in photoemission and optical properties of materials containing localized electrons. This is a tough problem. The full quantum-mechanical treatment is based on the solution of the Schrödinger equation for the many-body wavefunction, which is cumbersome and requires in practice some approximations. One simplified approach is given by Density Functional Theory (DFT) with the Kohn-Sham system of independent particles, but it is difficult to access properties other than the density and total energy. In this thesis we start from an in principle exact framework, the Green's functions. They are intermediate in complexity between DFT and the full wavefunction methods.For the removal or excitation of a localized electron one important point is to avoid self-interaction and self-screening. This is naturally achieved when one uses a generalized Coulomb interaction (Chap.3). Moreover, supposing that the localized electron has little overlap with the others, we can think that their interaction is classical. Then the main many-body effect is the reaction of the other electrons to the removal or excitation of the localized electron: this is screening of the hole or electron-hole pair by the other electrons. However, in many standard approximations in the Green's functions framework, such as GW or the cumulant expansion, screening appears in the linear response approximation. Instead, we can expect that the removal or excitation of a localized electron is a strong perturbation to the other electrons. Therefore, it could be that non-linear contributions to screening are important. How can we verify when this is true? And how can we include these effects? On the other hand, even in linear response one could do better than standard approximations, because the linear response screening itself is often calculated in the Random Phase Approximation (RPA). How much do things improve when one goes beyond the RPA but stays in linear response? We address these points in the thesis.Concerning the screening, in Chap.5 we first use a simple zero-dimensional model to study on one side, effects beyond the RPA within linear reponse and, on the other side, effects beyond linear response but staying within the RPA. Interestingly, we find that we have to treat both at the same time in order to find significant improvement. This means that we have to find promising ways to go beyond the RPA that are simple enough to allow us to go to the non-linear regime. Therefore we develop approximations based on perturbation theory and test some already existent ones in the model.Screening is expressed through the dielectric function, which gives us also directly absorption. This is another reason to study it. In order to be more realistic than the zero-dimensional model, in Chap.6 we study the dielectric function of a simple solid using localized Wannier functions. This allows us to highlight cancellations between self-energy and excitonic effects in the framework of Green's functions and from this derive a simple Kohn Sham exchange-correlation potential and kernel for Time-Dependent DFT (TDDFT).In Chap.7 we go back to the problem of non-linear screening and address the question: how can we make it appear explicitly in the full formulation? We show how to do this,and how to use the approximation of a localized electron in order to derive a cumulant Green's function beyond the standard linear response one. We propose two levels of approximations to evaluate the resulting expression in practice, and show some preliminary results. In both cases, TDDFT is used to describe screening.Since a combination of Green's functions and TDDFT seems to be a good strategy to simplify the many-body problem, Chap.8 contains some more considerations about possible combinations.

In this thesis I present a derivation of statistical mechanics starting from a closed, isolated quantum many body system. I first establish under what conditions we expect static equilibrium to emerge. It is found that the purity of the diagonal ensemble is a sufficient criteria for equilibration to occur and avoid short time recurrences. I next derive the usual ensembles of statistical mechanics using the principle of maximum entropy. These ensembles are then connected to the diagonal ensemble through the strong and the weak eigenstate thermalization hypothesis (ETH). Counter examples to ETH are discussed along with the process of scrambling. The thesis contains three contributed articles relevant to this introductory chapter studying early time relaxation, recurrences and scrambling. / Thesis / Doctor of Philosophy (PhD)

<p> It is known in the case of an electron gas that the Coulomb force between electrons is screened due to the density variation around the electrons. In a similar way the force acting between nucleons in nuclear matter may be appreciably different from the free nucleon-nucleon interactions. The main theme of this thesis is to examine the "Screening" effect of the nuclear force. To this end, first the dielectric formulation of the theory of an electron gas is reviewed. Relationship among the chain-diagram approximation, the random phase approximation and the dieledtric formulation is discussed in detail. These techniques are then applied to nuclear matter taking the one-pion exchange potential as an example. It is found that the screening effect on the nuclear force in nuclear matter is indeed quite appreciable. The validity of the approximations is discussed. </p> / Thesis / Master of Science (MSc)

We explore the dynamics of integrable and chaotic quantum many-body systems with a focus on universal structures known as caustics, which are a type of singularity categorized by catastrophe theory. Papers I and II study light cones in quantum spin chains, which we show are caustics and therefore inherits specific functional forms. For integrable systems, the edge of the cone is a fold catastrophe, making the wavefunction locally of Airy form. We also identify the cusp catastrophe in the XY model, thus the secondary light cone is a Pearcey function. Vortex pairs appear in the dynamics, are sensitive to phase transitions, and permit the extraction of critical scaling exponents. In paper II we use a Gaussian wavefront form to distinguish integrable and chaotic models. Writing the wavefront as exp[−m(x)(x − vt)2 + b(x)t], the scaling of coefficients m(x) and b(x) is the diagnostic. The local Airy function description in free models leads to a power-law ∼ x^{−n/3} scaling, while for the chaotic case the scaling is exponential ∼ e^{−cx}. In Paper III, we study the function Fn(t) = <(A(t)B)^n>, a generalization of the four-point out-of-time ordered correlator (OTOC) F2(t), for an integrable system and show that the function Fn(t) can be recast as the return amplitude of an effective time dependent chaotic system, exhibiting signals of chaos such as a positive Lyapunov exponent, spectral statistics consistent with random matrix theory, and relaxation. In Paper IV we perform a comprehensive investigation of caustics in many-body systems in (1+1)- and (2+1)-dimensional Fock space and time. We show how a hierarchy of caustics appear in the dynamics of many-body models, using two- and three-mode Bose-Hubbard models as guiding systems. We show that, in the case of the trimer, high dimensional caustics appear and are organized by the catastrophe X9. / Thesis / Doctor of Philosophy (PhD)

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