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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].

The purpose of this paper is to make a suggestion to deontic logic: Respect Hume\'s Law, the answer to the is-ought problem that says that all ought-talk is completely cut off from is-talk. Most deontic logicians have sought another solution: Namely, the solution that says that we can bridge the is-ought gap. Thus, a century\'s worth of research into these normative systems of logic has lead to many attempts at doing just that. At the same time, the field of deontic logic has come to be plagued with paradox. My argument essentially depends upon there being a substantive relation between this betrayal of Hume and the plethora of paradoxes that have appeared in two-adic (binary normative operator), one-adic (unary normative operator), and zero-adic (constant normative operator) deontic systems, expressed in the traditions of von Wright, Kripke, and Anderson, respectively. My suggestion has two motivations: First, to rid the philosophical literature of its puzzles and second, to give Hume\'s Law a proper formalization. Exploring the issues related to this project also points to the idea that maybe we should re-engineer (e.g., further generalize) our classical calculus, which might involve the adoption of many-valued logics somewhere down the line. / Master of Arts

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.

Spectral techniques in digital logic design have been known for more than thirty years. They have been used for Boolean function classification, disjoint decomposition, parallel and serial linear decomposition, spectral translation synthesis (extraction of linear pre- and post-filters), multiplexer synthesis, prime implicant extraction by spectral summation, threshold logic synthesis, estimation of logic complexity, testing, and state assignment. This dissertation resolves many important issues concerning the efficient application of spectral methods used in the computer-aided design of digital circuits. The main obstacles in these applications were, up to now, memory requirements for computer systems and lack of the possibility of calculating spectra directly from Boolean equations. By using the algorithms presented here these obstacles have been overcome. Moreover, the methods presented in this dissertation can be regarded as representatives of a whole family of methods and the approach presented can be easily adapted to other orthogonal transforms used in digital logic design. Algorithms are shown for Adding, Arithmetic, and Reed-Muller transforms. However, the main focus of this dissertation is on the efficient computer calculation of Rademacher-Walsh spectra of Boolean functions, since this particular ordering of Walsh transforms is most frequently used in digital logic design. A theory has been developed to calculate the Rademacher-Walsh transform from a cube array specification of incompletely specified Boolean functions. The importance of representing Boolean functions as arrays of disjoint ON- and DC- cubes has been pointed out, and an efficient new algorithm to generate disjoint cubes from non-disjoint ones has been designed. The transform algorithm makes use of the properties of an array of disjoint cubes and allows the determination of the spectral coefficients in an independent way. By such an approach each spectral coefficient can be calculated separately or all the coefficients can be calculated in parallel. These advantages are absent in the existing methods. The possibility of calculating only some coefficients is very important since there are many spectral methods in digital logic design for which the values of only a few selected coefficients are needed. Most of the current methods used in the spectral domain deal only with completely specified Boolean functions. On the other hand, all of the algorithms introduced here are valid, not only for completely specified Boolean functions, but for functions with don't cares. Don't care minterms are simply represented in the form of disjoint cubes. The links between spectral and classical methods used for designing digital circuits are described. The real meaning of spectral coefficients from Walsh and other orthogonal spectra in classical logic terms is shown. The relations presented here can be used for the calculation of different transforms. The methods are based on direct manipulations on Karnaugh maps. The conversion start with Karnaugh maps and generate the spectral coefficients. The spectral representation of multiple-valued input binary functions is proposed here for the first time. Such a representation is composed of a vector of Walsh transforms each vector is defined for one pair of the input variables of the function. The new representation has the advantage of being real-valued, thus having an easy interpretation. Since two types of codings of values of binary functions are used, two different spectra are introduced. The meaning of each spectral coefficient in classical logic terms is discussed. The mathematical relationships between the number of true, false, and don't care minterms and spectral coefficients are stated. These relationships can be used to calculate the spectral coefficients directly from the graphical representations of binary functions. Similarly to the spectral methods in classical logic design, the new spectral representation of binary functions can find applications in many problems of analysis, synthesis, and testing of circuits described by such functions. A new algorithm is shown that converts the disjoint cube representation of Boolean functions into fixed-polarity Generalized Reed-Muller Expansions (GRME). Since the known fast algorithm that generates the GRME, based on the factorization of the Reed-Muller transform matrix, always starts from the truth table (minterms) of a Boolean function, then the described method has advantages due to a smaller required computer memory. Moreover, for Boolean functions, described by only a few disjoint cubes, the method is much more efficient than the fast algorithm. By investigating a family of elementary second order matrices, new transforms of real vectors are introduced. When used for Boolean function transformations, these transforms are one-to-one mappings in a binary or ternary vector space. The concept of different polarities of the Arithmetic and Adding transforms has been introduced. New operations on matrices: horizontal, vertical, and vertical-horizontal joints (concatenations) are introduced. All previously known transforms, and those introduced in this dissertation can be characterized by two features: "ordering" and "polarity". When a transform exists for all possible polarities then it is said to be "generalized". For all of the transforms discussed, procedures are given for generalizing and defining for different orderings. The meaning of each spectral coefficient for a given transform is also presented in terms of standard logic gates. There exist six commonly used orderings of Walsh transforms: Hadamard, Rademacher, Kaczmarz, Paley, Cal-Sal, and X. By investigating the ways in which these known orderings are generated the author noticed that the same operations can be used to create some new orderings. The generation of two new Walsh transforms in Gray code orderings, from the straight binary code is shown. A recursive algorithm for the Gray code ordered Walsh transform is based on the new operator introduced in this presentation under the name of the "bi-symmetrical pseudo Kronecker product". The recursive algorithm is the basis for the flow diagram of a constant geometry fast Walsh transform in Gray code ordering. The algorithm is fast (N 10g2N additions/subtractions), computer efficient, and is implemented

In this paper I will argue that some of the contradictions in the Problem of the Many occur since we're dealing with two different types of concepts of a cloud, one that is 'common-sense', and one that is 'scientific'. I will borrow from Noam Chomsky’s distinction between common-sense concepts, which are understood intuitively and are usually human-centric, and scientific concepts, which are carefully constructed and aspire to be objective. The common-sense concept of a cloud has certain properties (such as having a sharp boundary) which contradict some found in the scientific concept (such as having a vague boundary), and this is the source of some contradictions found within the Problem of the Many. I will propose that in the future we should approach the problem with this in mind, and not treat it as if we're dealing with only one concept of a cloud, but instead, several.

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)