Many-body quantum dynamics is a challenging problem that has induced the development of many different computational techniques. One powerful technique is the multiconfiguration time-dependent Hartree-Fock (MCTDHF) method. This method allows proper consideration of electronic correlation with much less computational overhead compared to other similar methods. In this work, we present our implementation of the MCTDHF method on a non-uniform cylindrical grid. With the one-body limit of our code, we studied the controversial topic of tunneling delay, and showed that our results agree with one recent experiment while disagreeing with another. Using the fully correlated version of the code, we demonstrated the ability of MCTDHF to address correlation by calculating the ground state ionization energies of a few strongly correlated systems.

Orientador: Silvio Antonio Sachetto Vitiello / Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Física Gleb Wataghin / Made available in DSpace on 2018-08-24T13:18:32Z (GMT). No. of bitstreams: 1 Abreu_BrunoRicardide_M.pdf: 1928743 bytes, checksum: 32226a9b6b2fe6d0ce77dbb9efc50309 (MD5) Previous issue date: 2014 / Resumo: A matriz densidade é um objeto fundamental na mecânica estatística de sistemas de muitos corpos quânticos. Através dela pode ser encontrado o valor esperado de qualquer observável do sistema de interesse. Neste trabalho calculamos a matriz densidade a baixas temperaturas para sistemas de muitos corpos que interagem via um potencial de pares através de convolucões da matriz densidade a altas temperaturas, onde é possível utilizar aproximações semi-clássicas / Abstract: The density matrix is a fundamental object in statistical mechanics of quantum many-body systems. Through it the observed value of any observable of a quantum mechanical system of interest can be found. In this work we calculate the density matrix at low temperatures of manybody systems that interact through pairwise potentials using a convolution procedure of the density matrix at high temperatures, where is possible to apply semi-classical approximations / Mestrado / Física / Mestre em Física

Non-inferiority tests are well developed for randomized parallel group trials where the control and experimental groups are independent. However, these tests may not be appropriate for assessing non-inferiority in correlated one-to-many matched data. We propose a new statistical test that extends Farrington-Manning’s (FM) test to the case where many (≥1) control subjects are matched to each experimental subject. We conducted a Monte Carlo simulation study to compare the size and power of the proposed test with tests developed for clustered one-to-one matched pair data and tests based on generalized estimating equations (GEE). For various correlation patterns, the sizes of tests developed for clustered matched pair data and GEE-based tests are inflated when applied to the case where many control subjects are matched to each experimental subject. The size of the proposed test, on the other hand, is close to the nominal level for a variety of correlation patterns. There is a debate in the literature regarding whether or not statistical tests appropriate for independent samples can be used to assess the statistical significance of treatment effects in propensity-score matched studies. We used Monte Carlo simulations to examine the effect on assessing non-inferiority via risk difference when a method for independent samples (i.e. FM test) is used versus when a method for correlated matched samples is used in propensity-score one-to-many matched studies. If propensity-score matched samples are well-matched on baseline covariates and contain almost all of the experimental treated subjects, a method for correlated matched samples is preferable with respect to power and Type I error than a method for independent samples. Sometimes there are more experimental subjects to choose from for matching than control subjects. We conducted a Monte Carlo simulation study to compare the size and power of the previously mentioned tests when many (≥1) experimental subjects are matched to each control subject. In this case, the Nam-Kwon test for clustered data performs the best in controlling the type I error rate for a variety of correlation patterns. Therefore, the appropriate non-inferiority test to use for correlated matched data depends, in part, on the sample size allocation of subjects.

京都大学 / 0048 / 新制・課程博士 / 博士(理学) / 甲第21571号 / 理博第4478号 / 新制||理||1642(附属図書館) / 京都大学大学院理学研究科物理学・宇宙物理学専攻 / (主査)准教授 菅沼 秀夫, 教授 永江 知文, 教授 田中 貴浩 / 学位規則第4条第1項該当 / Doctor of Science / Kyoto University / DFAM

A defining feature of quantum many-body systems is the presence of entanglement among their constituents. Besides providing valuable insights on several physical properties, entanglement is also responsible for the computational complexity of simulating quantum systems with variational methods. This thesis explores several aspects of entanglement in many-body systems, with the primary goal of devising efficient approaches for the study of topological properties and quantum dynamics of lattice models. The first focus of this work is the development of variational wavefunctions inspired by artificial neural networks. These can efficiently encode long-range and extensive entanglement in their structure, as opposed to the case of tensor network states. This feature makes them promising tools for the study of topologically ordered phases, quantum critical states as well as dynamical properties of quantum systems. In this thesis, we characterize the representational power of a specific class of artificial neural network states, constructed from Boltzmann machines. First, we show that wavefunctions obtained from restricted Boltzmann machines can efficiently parametrize chiral topological phases, such as fractional quantum Hall states. We then turn our attention to deep Boltzmann machines. In this framework, we propose a new class of variational wavefunctions, coined generalized transfer matrix states, which encompass restricted Boltzmann machine and tensor network states. We investigate the entanglement properties of this ansatz, as well as its capability of representing physical states. Understanding how the entanglement properties of a system evolve in time is the second focus of this thesis. In this context, we first investigate the manifestation of topological properties in the unitary dynamics of systems after a quench, using the degeneracy of the entanglement spectrum as a possible signature. We then analyze the phenomenon of entanglement growth, which limits to short timescales the applicability of tensor network methods in out-of-equilibrium problems. We investigate whether these limitations can be overcome by exploiting the dependence of entanglement entropies on the chosen computational basis. Specifically, we study how the spreading of quantum correlations can be contained by means of time-dependent basis rotations of the state, using exact diagonalization to simulate its dynamics after a quench. Going beyond the case of sudden quenches, we then show how, in certain weakly interacting problems, the asymptotic value of the entanglement entropy can be tuned by modifying the velocity at which the parameters in the Hamiltonian are changed. This enables the simulation of longer timescales using tensor network approaches. We present preliminary results obtained with matrix product states methods, with the goal of studying how equilibration affects the transport properties of interacting systems at long times.

‘If-then’ rule sets are one of the most expressive and human-readable knowledge representations. This thesis deals with optimization and analysis of decision and inhibitory rules for decision tables with many-valued decisions. The most important areas of applications are knowledge extraction and representation. The benefit of considering inhibitory rules is connected with the fact that in some situations they can describe more knowledge than the decision ones. Decision tables with many-valued decisions arise in combinatorial optimization, computational geometry, fault diagnosis, and especially under the processing of data sets. In this thesis, various examples of real-life problems are considered which help to understand the motivation of the investigation. We extend relatively simple results obtained earlier for decision rules over decision tables with many-valued decisions to the case of inhibitory rules. The behavior of Shannon functions (which characterize complexity of rule systems) is studied for finite and infinite information systems, for global and local approaches, and for decision and inhibitory rules. The extensions of dynamic programming for the study of decision rules over decision tables with single-valued decisions are generalized to the case of decision tables with many-valued decisions. These results are also extended to the case of inhibitory rules. As a result, we have algorithms (i) for multi-stage optimization of rules relative to such criteria as length or coverage, (ii) for counting the number of optimal rules, (iii) for construction of Pareto optimal points for bi-criteria optimization problems, (iv) for construction of graphs describing relationships between two cost functions, and (v) for construction of graphs describing relationships between cost and accuracy of rules. The applications of created tools include comparison (based on information about Pareto optimal points) of greedy heuristics for bi-criteria optimization of rules, and construction (based on multi-stage optimization of rules) of relatively short systems of rules that can be used for knowledge representation.

Au sein de physique des systèmes quantiques désordonnés, le domaine des atomes ultra-froids est en pleine croissance. En l’occurrence, l'étude de la relation entre la localisation et les interactions a permis de découvrir la richesse de la physique de la localisation à N-corps. Ce phénomène remarquable fournit un mécanisme pour la brisure de l'ergodicité dans les systèmes quantiques isolés et désordonnés. Plusieurs questions ont été évoquées après cette découverte, comme la possibilité d'une transition fluide-isolant à température finie. Dans cette thèse, j'étudie la localisation à N-corps dans le contexte de bosons désordonnés à deux dimensions. Dans la première partie, je présente l'étude d'un gaz interactif de Bose bidimensionnel dans un potentiel aléatoire à température finie. Le système présente deux transitions à température finie: la transition de localisation à N-corps entre fluide et isolant, et la transition de Berezinskii-Kosterlitz-Thouless entre superfluide algébrique et fluide. J'examine ensuite l'influence de la troncature de la distribution d'énergie dû au piégeage, un phénomène générique dans le cadre du refroidissement d'atomes ultra-froids. Finalement, je conclus en discutant la stabilité de la phase isolante dans des systèmes définis sur un continuum. / The study of the interplay between localization and interactions in disordered quantum systems led to the discovery of the interesting physics of many-body localization (MBL). This remarkable phenomenon provides a generic mechanism for the breaking of ergodicity in quantum isolated systems, and has stimulated several questions such as the possibility of a finite-temperature fluid-insulator transition. At the same time, the domain of ultracold interacting atoms is a rapidly growing field in the physics of disordered quantum systems. In this thesis, we study many-body localization in the context of two-dimensional disordered ultracold bosons. After reviewing some importance concepts, we present a study of the phase diagram of a two-dimensional weakly interacting Bose gas in a random potential at finite temperatures. The system undergoes two finite-temperature transitions: the MBL transition from normal fluid to insulator and the Berezinskii-Kosterlitz-Thouless transition from algebraic superfluid to normal fluid. At T=0, we show the existence of a tricritical point where the three phases coexist. We also discuss the influence of the truncation of the energy distribution function at the trap barrier, a generic phenomenon for ultracold atoms. The truncation limits the growth of the localization length with energy and, in contrast to the thermodynamic limit, the insulator phase is present at any temperature. Finally, we conclude by discussing the stability of the insulating phase with respect to highly energetic particles in systems defined on a continuum.

In 1942, Paul C. Rosenbloom put out a definition of a Post algebra after Emil L. Post published a collection of systems of many–valued logic. Post algebras became easier to handle following George Epstein’s alternative definition. As conceived by Rosenbloom, Post algebras were meant to capture the algebraic properties of Post’s systems; this fact was not verified by Rosenbloom nor Epstein and has been assumed by others in the field. In this thesis, the long–awaited demonstration of this oft–asserted assertion is given. After an elemental history of many–valued logic and a review of basic Classical Propositional Logic, the systems given by Post are introduced. The definition of a Post algebra according to Rosenbloom together with an examination of the meaning of its notation in the context of Post’s systems are given. Epstein’s definition of a Post algebra follows the necessary concepts from lattice theory, making it possible to prove that Post’s systems of many–valued logic do in fact form a Post algebra.

Es wird eine sprachliche Erweiterung der Aussagenlogik vorgeschlagen. Es handelt sich um eine Art von schwacher Negation ('disbelief'). Eine entsprechende Logik wird entwickelt. Diese wird semantisch charakterisiert. Weiterhin wird auf Schwierigkeiten hingewiesen, die bei der Axiomatisierung auftreten werden.

In this thesis, we focus attention on the effects of disorder in closed interacting quantum systems that give rise to a many-body localization (MBL) transition between an ergodic phase and a many-body localized phase. This transition is not a conventional one, since it takes place at any finite energy density and can neither be described by thermodynamics nor conventional statistical mechanics. We explain why systems experiencing such an MBL transition can be regarded as generic in many ways, we do so by discussing many of their spectral properties and by giving a detailed account of their manifestation in the nonequilibrium dynamics and long-time behavior. Surprisingly, a wide variety of MBL systems consistently reflect strikingly similar characteristic effects in each side of the MBL transition. This is backed by myriads of numerical and experimental observations which in turn can be partially explained by theories developed in the past decade. However, some mechanisms behind the ergodic side of the MBL transition and the nature of the MBL transition itself remain elusive. These, as well as the lack of an accurate description of the nonergodic character of the steady states of such systems, have been some of the issues for active research and speculation by scholars that need to be timely addressed. In the following, we describe our modest contributions at bridging the gap of understanding of some of the issues exposed above. On the one hand, reduced density matrices are central objects for the description of the relaxation of local observables in closed quantum many-body systems, and on the other, quench protocols are experimentally relevant procedures. In the first part of this thesis we study the long-time behavior of the one-particle density matrix (OPDM) occupation spectrum after a quench. It was shown that, in the many-body localized phase (which can be understood in terms of localized quasiparticles), the OPDM occupation spectrum in eigenstates shows a zero-temperature Fermi liquid-like discontinuity at any finite energy density. In this thesis we show that in the steady state reached at long times after a global quench from a perfect density-wave state, the discontinuity in the OPDM occupation spectrum is absent, reminiscent of a Fermi liquid at a finite temperature, while the full occupation function remains strongly nonthermal. We discuss how one can understand this as a consequence of the local structure of the density-wave state and the resulting partial occupation of quasiparticles. We further show how these partial occupations can be controlled by tuning the structure of initial state and described by an effective temperature. Another part of this thesis was devoted to the study of dynamics on the ergodic side of the transition in periodically driven systems in the absence of global conservation laws. Most numerical studies in this context were done in models with conserved quantities (e.g., energy and/or particle number) which could account for the reduction of the overall complexity of the problem, while in this thesis, we use a numerical technique based on the fast Walsh-Hadamard transform that allows us to perform an exact time evolution for large systems and long times. As in models with conserved quantities, we observe a slowing down of the dynamics as the transition into the many-body localized phase is approached. This is reflected in anomalous behavior of the energy absorption of the system, as well as consistent with a subballistic spread of entanglement and a stretched-exponential decay of an autocorrelation function, with their associated exponents reflecting slow dynamics near the transition for a fixed system size. However, with access to larger system sizes, we observe a clear flow of the exponents towards faster dynamics and cannot rule out that the slow dynamics is a finite-size effect. Furthermore, we observe examples of nonmonotonic dependence of the exponents with time, with the dynamics initially slowing down but accelerating again at larger times, which could be consistent with the slow dynamics being a crossover phenomenon with a localized critical point. In addition, we observe no difference between the typical and average value of the autocorrelation function and therefore our results are inconsistent with the phenomenological explanation of the anomalous behavior based on Griffiths effects. In the last part of this thesis, we study dynamics in the ergodic phase relating to two main quantum information measures: One is the entanglement entropy, which is an intrinsic property of the wave function and generated by the time evolution operator, while the other is the operator entanglement entropy of the time evolution operator, which quantifies the complexity of the latter. It is known that generic quantum many-body systems typically show a linear growth of the entanglement entropy growth after a quench from a product state. In this thesis we show that there is a robust correspondence between the operator entanglement entropy of the time evolution operator and the entanglement entropy growth of typical product states, whereas special product states, e.g., $\sigma_z$ basis states, may exhibit faster entanglement production. We base our analysis on numerical simulations of a static and a periodically driven quantum spin chain in the presence of a disordered magnetic field, showing that both the wave function and operator entanglement entropies exhibit a power-law growth with the same disorder-dependent exponent. With this, we clarify the discrepancy between the exponents observed in previous results. Our results provide further evidence for slow information spreading on the ergodic side of the many-body localization transition in the absence of conservation laws. / In dieser Dissertation setzen wir uns mit dem Effekt von Unordnung auf geschlossene wechselwirkende Quantensysteme auseinander. Unordnung kann einen Übergang von einer ergodischen in eine lokalisierte Phase induzieren, eine sogenannte Vielteilchenlokalisierung oder Many body localization (MBL). Dieser Phasenübergang ist alles andere als konventionell: Er kann weder durch Thermodynamik noch durch klassische statistische Mechanik beschrieben werden. Wir erklären, warum Systeme, die solch einen MBL Übergang aufweisen, in vielerlei Hinsicht als generisch angesehen werden können. Dazu diskutieren wir die spektralen Eigenschaften, die Nichtgleichgewichtsdynamik und das Langzeitverhalten. Erstaunlicherweise weist eine große Vielfalt verschiedener MBL Systeme auf beiden Seiten des MBL Übergangs mit großer Konsistenz ähnliche Charakteristiken auf. Dies wird durch unzählige numerische und experimentelle Beobachtungen unterstützt, die wiederum zumindest teilweise durch theoretische Arbeiten aus dem letzten Jahrzehnt erklärt werden können. Trotzdem bleiben manche Mechanismen auf der ergodischen Seite des MBL Übergangs und die Art des MBL Übergangs weiterhin im Verborgenen. Zusammen mit der fehlenden akkuraten Beschreibung des nicht-ergodischen Charakters der stationären Zustände dieser Systeme sind diese Probleme im derzeitigen Fokus der Forschung, wobei es eine Vielzahl fundierter Vermutungen gibt, die diese Phänomene erklären. Im Folgenden beschreiben wir unseren Beitrag wie diese oben gelisteten Probleme überwunden werden können. Reduzierte Dichteoperatoren sind zentrale Objekte, um die Relaxation von lokalen Observablen in geschlossenen Quantenvielkörpersystemen zu beschreiben und sogenannte Quenches, also die plötzliche Änderung einiger systemrelevanter Parameter, ähnlich wie beim Abschrecken mit Wasser oder Luft, sind experimentell relevante Vorgänge. Im ersten Teil dieser Arbeit untersuchen wir das Langzeitverhalten des Besetzungsspektrums des Einteilchendichteoperators (one-particle density matrix, OPDM) nach solch einem Quench. Wie zuvor gezeigt wurde, weist das OPDM Besetzungsspektrum in der MBL Phase (die im Sinne von lokalisierten Quasiteilchen verstanden werden kann) für alle endlichen Energiedichten eine Diskontinuität auf, ähnlich wie in Fermi-Flüssigkeiten. In dieser Arbeit zeigen wir, dass diese Diskontinuität in stationären Zuständen, die von perfekten Dichtewellen ausgehend nach langer Zeit nach einem globalen Quench erreicht werden, abwesend ist, ähnlich wie in einer Fermi-Flüssigkeit bei einer endlichen Temperatur, während die gesamte Besetzungsfunktion stark nicht-thermal bleibt. Wir diskutieren, wie man dies als Konsequenz der lokalen Struktur des Dichtewellenzustands und der daraus folgenden teilweisen Besetzung der Quasiteilchen verstehen kann. Wir zeigen außerdem, wie die teilweise Besetzung durch Änderung der Struktur des Ausgangszustands kontrolliert und durch eine effektive Temperatur beschrieben werden kann. Im nächsten Teil dieser Arbeit untersuchen wir die Dynamik der ergodischen Seite des MBL Übergangs in periodisch getriebenen Systemen ohne globale Erhaltungsgrößen. Die meisten bisherigen in diesem Zusammenhang vorgenommenen numerischen Untersuchungen wurden in Modellen mit Erhaltungsgrößen (wie Energie und/oder Teilchenzahl) durchgeführt, was an der Reduzierung der Komplexität des Problems liegen mag. In dieser Arbeit nutzen wir hingegen eine numerische Methode, die auf einer schnellen Walsh-Hadamard Transformation beruht, was uns ermöglicht, eine exakte Zeitentwicklung für lange Zeiten und große Systeme vorzunehmen. Wie in Modellen mit Erhaltungsgrößen beobachten wir eine Verlangsamung der Dynamik, wenn wir uns dem Übergangspunkt zu der MBL Phase nähern. Dies macht sich in einem ungewöhnlichen Verhalten der Energieabsorption des Systems bemerkbar, was mit einer unterballistischen Ausbreitung der Verschränkung und einem gedehnt-exponentiellen Abklingen der Autokorrelationsfunktion im Einklang steht, wobei die zugehörigen Exponenten die verlangsamte Dynamik für fixe Systemgrößen widerspiegeln. Durch den Zugang zu größeren Systemen können wir jedoch einen deutlichen Fluss der Exponenten Richtung schnellerer Dynamik feststellen und daher nicht ausschließen, dass die verlangsamte Dynamik durch die endlichen Systemgrößen hervorgerufen wird (ein sogenannter finite size effect). Des weiteren finden wir Beispiele für eine nicht-monotone Zeitabhängigkeit der Exponenten, wobei die Dynamik sich zunächst verlangsamt, bevor sie zu späteren Zeiten wieder beschleunigt. Dies könnte mit der Betrachtung der verlangsamten Dynamik als Crossover-Phänomen mit einem lokalisierten kritischen Punkt vereinbar sein. Außerdem können wir keinen Unterschied zwischen dem geometrischen und arithmetischen Mittel der Autokorrelationsfunktion feststellen, sodass unsere Ergebnisse der phänomenologischen Erklärung des ungewöhnlichen Verhaltens, die auf Griffiths-Effekten beruht, widersprechen. Im letzten Teil der Dissertation widmen wir der Dynamik in der ergodischen Phase und verknüpfen zwei zentrale Größen der Quanteninformation: die Verschränkungsentropie, eine der Wellenfunktion intrinsische Größe, die aus dem Zeitentwicklungsoperator generiert werden kann, und der Operatorverschränkungsentropie des Zeitentwicklungsoperators, die die Komplexität des Operators quantifiziert. In generischen Quantenvielkörpersystemen wächst die Verschränkungsentropie nach einem Quench aus einem Produktzustand typischerweise linear. In dieser Arbeit zeigen wir, dass es eine belastbaren Übereinstimmung zwischen der Operatorverschränkungsentropie des Zeitentwicklungsoperators und der Verschränkungsentropie typischer Produktzustände gibt, wobei bestimmte Produktzustände, z.B. $\sigma_z$-Basiszustände, eine schnellere Verschränkungsproduktion aufweisen können. Unsere Analyse basiert auf numerischen Simulationen von statischen und periodisch getriebenen Quanten-Spinketten in einem ungeordneten Magnetfeld. Sowohl die Verschränkungsentropie der Wellenfunktion als auch die Operatorverschränkungsentropie wächst einem Potenzgesetz folgend mit den selben unordnungsabhängigen Exponenten. Damit schaffen wir Klarheit bezüglich der Unstimmigkeiten der Exponenten in den vorherigen Ergebnissen. Unsere Resultate geben außerdem Hinweise auf eine verlangsamte Informationsausbreitung auf der ergodischen Seite des MBL Übergangs ohne Erhaltungsgrößen.