<div><div><div><p>When confined to a finite, two-dimensional area and exposed to a strong magnetic field, electrons exhibit a complicated, highly correlated quantum behavior known as the quantum Hall effect. This dissertation consists of finite size numerical investigations of this effect. One line of study develops treatment of the fractional quantum Hall effect using the hyperspherical method, in conjunction with applications to the few-body quantum Hall systems, e.g., highly-controlled atomic systems. Another line of research fully utilizes the developed numerical techniques to study on the platform of finite size fractional quantum Hall states the bulk-edge correspondence principle, which is universal for phases in topological orders. It has been demonstrated that the eigenstates associated with the entanglement spectrum reveal more information about the ground state than the spectrum alone.</p></div></div></div>

La spectroscopie d'intrication, initialement introduite par Li et Haldane dans le contexte de l'effet Hall quantique fractionnaire, a suscité un large éventail de travaux. Le spectre d'intrication est le spectre de la matrice de densité réduite, quand on partitionne le système en deux. Pour de nombreux systèmes quantiques, il révèle une caractéristique unique : calculé uniquement à partir de la fonction d'onde de l'état fondamental, le spectre d'intrication donne accès à la physique des excitations de bord. Dans ce manuscrit, nous donnons un apercu de la spectroscopie d'intrication. Nous introduisons les concepts de base dans le cas des chaînes de spins quantiques. Nous présentons une étude approfondie des spectres d'intrication appliqués aux phases de l'effet Hall quantique fractionnaire, montrant quel type d'information est encodé dans l'état fondamental et comment les différentes facons de partitionner le système permettent de sonder différents types d'excitation. Comme application pratique de cette technique, nous discutons de la manière dont cette technique peut aider à faire la distinction entre les différentes phases qui émergent dans les isolants de Chern en interaction forte.

Phase transitions are fundamental phenomena in (quantum) many-body systems. They are associated with changes in the macroscopic physical properties of the system in response to the alteration in the conditions controlled by one or more parameters, like temperature or coupling constants. Quantum phase transitions are particularly intriguing as they reveal new insights into the fundamental nature of matter and the laws of physics. The study of phase transitions in such systems is crucial in aiding our understanding of how materials behave in extreme conditions, which are difficult to replicate in laboratory, and also the behavior of exotic states of matter with unique and potentially useful properties like superconductors and superfluids. Moreover, this understanding has other practical applications and can lead to the development of new materials with specific properties or more efficient technologies, such as quantum computers. Hence, detecting the transition point from one phase of matter to another and constructing the corresponding phase diagram is of great importance for examining many-body systems and predicting their response to external perturbations. Traditionally, phase transitions have been identified either through analytical methods like mean field theory or numerical simulations. The pinpointing of the critical value normally involves the measure of specific quantities such as local observables, correlation functions, energy gaps, etc. reflecting the changes in the physics through the transition. However, the latter approach requires prior knowledge of the system to calculate the order parameter of the transition, which is uniquely associated to its universality class. Recently, another method has gained more and more attention in the physics community. By using raw and very general representative data of the system, one can resort to machine learning techniques to distinguish among patterns within the data belonging to different phases. The relevance of these techniques is rooted in the ability of a properly trained machine to efficiently process complex data for the sake of pursuing classification tasks, pattern recognition, generating brand new data and even developing decision processes. The aim of this thesis is to explore phase transitions from this new and promising data-centric perspective. On the one hand, our work is focused on the developement of new machine learning architectures using state-of-the-art and interpretable models. On the other hand, we are interested in the study of the various possible data which can be fed to the artificial intelligence model for the mapping of a quantum many-body system phase diagram. Our analysis is supported by numerical examples obtained via matrix-product-states (MPS) simulations for several one-dimensional zero-temperature systems on a lattice such as the XXZ model, the Extended Bose-Hubbard model (EBH) and the two-species Bose Hubbard model (BH2S). In Part I, we provide a general introduction to the background concepts for the understanding of the physics and the numerical methods used for the simulations and the analysis with deep learning. In Part II, we first present the models of the quantum many-body systems that we study. Then, we discuss the machine learning protocol to identify phase transitions, namely anomaly detection technique, that involves the training of a model on a dataset of normal behavior and use it to recognize deviations from this behavior on test data. The latter can be applied for our purpose by training in a known phase so that, at test-time, all the other phases of the system are marked as anomalies. Our method is based on Generative Adversarial Networks (GANs) and improves the networks adopted by the previous works in the literature for the anomaly detection scheme taking advantage of the adversarial training procedure. Specifically, we train the GAN on a dataset composed of bipartite entanglement spectra (ES) obtained from Tensor Network simulations for the three aforementioned quantum systems. We focus our study on the detection of the elusive Berezinskii-Kosterlitz-Thouless (BKT) transition that have been object of intense theoretical and experimental studies since its first prediction for the classical two-dimensional XY model. The absence of an explicit symmetry breaking and its gappless-to-gapped nature which characterize such a transition make the latter very subtle to be detected, hence providing a challenging testing ground for the machine-driven method. We train the GAN architecture on the ES data in the gapless side of BKT transition and we show that the GAN is able to automatically distinguish between data from the same phase and beyond the BKT. The protocol that we develop is not supposed to become a substitute to the traditional methods for the phase transitions detection but allows to obtain a qualitative map of a phase diagram with almost no prior knowledge about the nature and the arrangement of the phases -- in this sense we refer to it as agnostic -- in an automatic fashion. Furthermore, it is very general and it can be applied in principle to all kind of representative data of the system coming both from experiments and numerics, as long as they have different patterns (even hidden to the eye) in different phases. Since the kind of data is crucially linked with the success of the detection, together with the ES we investigate another candidate: the probability density function (PDF) of a globally U(1) conserved charge in an extensive sub-portion of the system. The full PDF is one of the possible reductions of the ES which is known to exhibit relations and degeneracies reflecting very peculiar aspects of the physics and the symmetries of the system. Its patterns are often used to tell different kinds of phases apart and embed information about non-local quantum correlations. However, the PDF is measurable, e.g. in quantum gas microscopes experiments, and it is quite general so that it can be considered not only in the cases of the study but also in other systems with different symmetries and dimensionalities. Both the ES and the PDF can be extracted from the simulation of the ground state by dividing the one-dimensional chain into two complementary subportions. For the EBH we calculate the PDF of the bosonic occupation number in a wide range of values of the couplings and we are able to reproduce the very rich phase diagram containing several phases (superfluid, Mott insulator, charge density wave, phase separation of supersolid and superfluid and the topological Haldane insulator) just with an educated gaussian fit of the PDF. Even without resorting to machine learning, this analysis is instrumental to show the importance of the experimentally accessible PDF for the task. Moreover, we highlight some of its properties according to the gapless and gapped nature of the ground state which require a further investigation and extension beyond zero-temperature regimes and one-dimensional systems. The last chapter of the results contains the description of another architecture, namely the Concrete Autoencoder (CAE) which can be used for detecting phase transitions with the anomaly detection scheme while being able to automatically learn what the most relevant components of the input data are. We show that the CAE can recognize the important eigenvalues out of the entire ES for the EBH model in order to characterize the gapless phase. Therefore the latter architecture can be used to provide not only a more compact version of the input data (dimensionality reduction) -- which can improve the training -- but also some meaningful insights in the spirit of machine learning interpretability. In conclusion, in this thesis we describe two advances in the solution to the problem of phase recognition in quantum many-body systems. On one side, we improve the literature standard anomaly detection protocol for an automatic and agnostic identification of the phases by employing the GAN network. Moreover, we implement and test an explainable model which can make the interpretation of the results easier. On the other side we put the focus on the PDF as a new candidate quantity for the scope of discerning phases of matter. We show that it contains a lot of information about the many-body state being very general and experimentally accessible.