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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.

Au cours de cette thèse, nous avons étudié le comportement critique de chaînes de spins quantiques en présence de couplages désordonnés ou répartis de manière apériodique. Il est bien établi que le comportement critique des chaînes de spins quantiques d’Ising et de Potts est gouverné par le même point fixe de désordre infini. Nous avons implémenté́ une version numérique de la technique de renormalisation de désordre infini (SDRG) afin de tester cette prédiction. Dans un second temps, nous avons étudié la chaîne quantique d’Ashkin-Teller désordonnée par renormalisation de la matrice densité́ (DMRG). Nous confirmons le diagramme de phase précédemment proposé en déterminant la position des pics du temps d’autocorrélation intégré des corrélations spin-spin et polarisation-polarisation ainsi que ceux des fluctuations de l’aimantation et de la polarisation. Enfin, l’existence d’une double phase de Griffiths est confirmée par une étude détaillée de la décroissance des fonctions d’autocorrélation en dehors des lignes critiques. Comme attendu, l’exposant dynamique diverge à l’approche de ces lignes. Dans le cas apériodique, nous avons étudié les chaînes quantiques d’Ising et de Potts. En utilisant la méthode SDRG, nous avons confirmé les résultats connus pour la chaîne d’Ising et proposé des estimations de la dimension d’échelle magnétique. Dans le cas du modèle de Potts à q états, nous avons estimé l’exposant magnétique et observé qu’il était indépendant du nombre d’états q pour toutes les séquences dont l’exposant de divagation est nul. Toutefois, nous montrons que l’exposant dynamique est fini et augmente avec le nombre d’états q. En revanche, pour la séquence de Rudin-Shapiro, les résultats sont compatibles avec un point fixe de désordre infini et donc un exposant dynamique infini. / In the present thesis, the critical and off-critical behaviors of quantum spin chains in presence of a random or an aperiodic perturbation of the couplings is studied. The critical behavior of the Ising and Potts random quantum chains is known to be governed by the same Infinite-Disorder Fixed Point. We have implemented a numerical version of the Strong-Disorder Renormalization Group (SDRG) to test this prediction. We then studied the quantum random Ashkin-Teller chain by Density Matrix Renormalization Group. The phase diagram, previously obtained by SDRG, is confirmed by estimating the location of the peaks of the integrated autocorrelation times of both the spin-spin and polarization-polarization autocorrelation functions and of the disorder fluctuations of magnetization and polarization. Finally, the existence of a double-Griffiths phase is shown by a detailed study of the decay of the off-critical autocorrelation functions. As expected, a divergence of the dynamical exponent is observed along the two transition lines. In the aperiodic case, we studied both the Ising and Potts quantum chains. Using numerical SDRG, we confirmed the known analytical results for the Ising chains and proposed a new estimate of the magnetic scaling dimension.For the quantum q-state Potts chain, we estimated the magnetic scaling dimension for various aperiodic sequences and showed that it is independent of q for all sequences with a vanishing wandering exponent. However, we observed that the dynamical exponent is finite and increases with the number of states q. In contrast, for the Rudin-Shapiro sequence, the results are compatible with an Infinite-Disorder Fixed Point with a diverging dynamical exponent, equipe de renormalization

In this thesis, we investigate the entanglement and magnetic properties of frustrated spin systems and correlated electronic properties of conjugated carbon systems. In chapter 1, we present different approaches to solve the time-independent, nonrelativistic Schr¨odinger equation for a many-body system. We start with the full non-relativistic Hamiltonian of a multi nuclear system to describe the Born - Oppenheimer approximation which allows the study of electronic Hamiltonian which treats nuclear positions parametrically. We then also describe ab initio techniques such as the Hartree-Fock Method and density functional theories. We then introduce model Hamiltonians for strongly correlated systems such as the Hubbard, Pariser-Parr-Pople and Heisenberg models, and show how they result from the noninteracting one-band tight-binding model. In chapter 2, we discuss various numerical techniques like the exact diagonalization methods and density matrix renormalization group (DMRG) method. We also discuss quantum entanglement and the success of DMRG which can be attributed to the area law of entanglement entropy. In chapter 3, we study here different regions in phase diagrams of the spin-1/2, spin-1 and spin-3/2 one-dimensional antiferromagnetic Heisenberg systems with nearest-neighbor (J1) and next-nearest-neighbor (J2) interactions and dimerization (d ). Frustration arises for specific relative signs of the interactions J1 and J2. In particular, we analyze the behavior of the bipartite entanglement entropy and fidelity at the gapless to gapped phase transitions and across the lines separating different phases in the J2−d plane. All the calculations in this work are based on exact diagonalizations of finite systems. In chapter 4, we study Heisenberg spin-1/2 and spin-1 chains with alternating ferromagnetic (JF 1 ) and antiferromagnetic (JA 1 ) nearest-neighbor interactions and a ferromagnetic next-nearest-neighbor interaction (JF 2 ). In this model frustration is present due to non-zero JF 2 . The model with site spin s behaves like a Haldane spin chain with site spin 2s in the limit of vanishing JF 2 and large JF 1 /JA 1 . We show that the exact ground state of the model can be found along a line in the parameter space. For fixed JF 1 , the phase diagram in the space of JA 1 −JF 2 is determined using numerical techniques complemented by analytical calculations. A number of quantities, including the structure factor, energy gap, entanglement entropy and zero temperature magnetization, are studied to understand the complete phase diagram. An interesting and potentially important feature of this model is that it can exhibit a macroscopic magnetization jump in the presence of a magnetic field; we study this using an effective Hamiltonian. In chapter 5, we study correlated electronic properties of zigzag and armchair fused naphthalenes and polyperylene systems in the presence of long-range electronelectron interactions. We find that the ground state of zigzag fused naphthalene system is a higher spin state, while the ground state of armchair fused naphthalene is a singlet. The spin gap of polyperylene is unusually small and the ground state is a singlet. Our calculations of optical gap and two-photon gap suggest that polyperylene should exhibit fluorescence. From the charge gap calculation, we predict that in zigzag fused naphthalene and polyperylene systems, excitons are weakly binding. Peierls type of distortion is negligible in zigzag fused naphthalene and polyperylene systems, however, in armchair fused naphthalene system, interior bonds have tendency to distort in low-lying excited states. In chapter 6, we study the ground state spin of the Heisenberg spin-1/2 nearestneighboring antiferromagnetic exchange models of systems with fused odd member rings. In particular, we compute the ground state spin of fused three and five membered rings as well as fused five membered rings. In the thermodynamic limit, the ground state of the fused three and five membered system is a higher spin state, while fused five membered system shows a singlet ground state, for all system sizes.

Cold-atom quantum simulators offer unique possibilities to prepare, manipulate, and probe quantum many-body systems. However, despite the high level of control in modern experiments, not all observables of interest are easily accessible. This thesis aims at establishing protocols to measure currently elusive static and dynamic properties of quantum systems. The experimental feasibility of these schemes is illustrated by means of numerical simulations for relevant applications in many-body physics and quantum simulation. In particular, we introduce a general method for measuring dynamical correlations based on non-Hermitian linear response. This enables unbiased tests of the famous fluctuation-dissipation relation as a probe of thermalization in isolated quantum systems. Furthermore, we develop ancilla-based techniques for the measurement of currents and current correlations, permitting the characterization of strongly correlated quantum matter. Another application is geared towards revealing signatures of supersolidity in spin-orbit-coupled Bose gases by exciting the relevant Goldstone modes. Finally, we explore a scenario for quantum-simulating post-inflationary reheating dynamics by parametrically driving a Bose gas into the regime of universal far-from-equilibrium dynamics. The presented protocols also apply to other analog quantum simulation platforms and thus open up promising applications in the field of quantum science and technology. / I simulatori quantistici ad atomi freddi offrono possibilità uniche per preparare, manipolare e sondare sistemi quantistici a molti corpi. Tuttavia, nonostante l'alto livello di controllo raggiunto negli esperimenti moderni, non tutte le osservabili di interesse sono facilmente accessibili. Lo scopo di questa tesi è quello di stabilire protocolli per misurare delle proprietà statiche e dinamiche dei sistemi quantistici attualmente inaccessibili. La fattibilità sperimentale di questi schemi è illustrata mediante simulazioni numeriche per applicazioni rilevanti nella fisica a molti corpi e nella simulazione quantistica. In particolare, introduciamo un metodo generale per misurare le correlazioni dinamiche basato su una risposta lineare non hermitiana. Ciò consente test imparziali della famosa relazione fluttuazione-dissipazione come sonda di termalizzazione in sistemi quantistici isolati. Inoltre, sviluppiamo tecniche basate su ancilla per la misura di correnti e correlazioni di corrente, consentendo la caratterizzazione della materia quantistica fortemente correlata. Un'altra applicazione è orientata a rivelare l'impronta della supersolidità nei gas Bose con accoppiamento spin-orbita eccitando il corrispondente modo di Goldstone. Infine, esploriamo uno scenario per la simulazione quantistica della dinamica di riscaldamento post-inflazione modulando parametricamente un gas Bose e portandolo nel regime della dinamica universale lontana dall'equilibrio. I protocolli presentati si applicano anche ad altre piattaforme di simulazione quantistica analogica e aprono quindi applicazioni promettenti nel campo della scienza e della tecnologia quantistica. / Quantensimulatoren auf Basis ultrakalter Atome eröffnen einzigartige Möglichkeiten zur Präparation, Manipulation und Untersuchung von Quanten-Vielteilchen-Systemen. Trotz des hohen Maßes an Kontrolle in modernen Experimenten sind jedoch nicht alle interessanten Observablen auf einfache Weise zugänglich. Ziel dieser Arbeit ist es, Protokolle zur Messung aktuell nur schwer erfassbarer statischer und dynamischer Eigenschaften von Quantensystemen zu etablieren. Die experimentelle Realisierbarkeit dieser Verfahren wird durch numerische Simulationen anhand relevanter Anwendungen in der Vielteilchenphysik und Quantensimulation veranschaulicht. Insbesondere wird eine allgemeine Methode zur Messung dynamischer Korrelationen basierend auf der linearen Antwort auf nicht-hermitesche Störungen vorgestellt. Diese ermöglicht unabhängige Tests des berühmten Fluktuations-Dissipations-Theorems als Indikator der Thermalisierung isolierter Quantensysteme. Darüber hinaus werden Verfahren zur Messung von Strömen und Strom-Korrelationen mittels Kopplung an einen Hilfszustand entwickelt, welche die Charakterisierung stark korrelierter Quantenmaterie erlauben. Eine weitere Anwendung zielt auf die Enthüllung spezifischer Merkmale von Supersolidität in Spin-Bahn-gekoppelten Bose-Einstein-Kondensaten ab, indem die relevanten Goldstone-Moden angeregt werden. Schließlich wird ein Szenario zur Quantensimulation post-inflationärer Thermalisierungsdynamik durch die parametrische Anregung eines Bose-Gases in das Regime universeller Dynamik fern des Gleichgewichts erschlossen. Die dargestellten Protokolle lassen sich auch auf andere Plattformen für analoge Quantensimulation übertragen und eröffnen damit vielversprechende Anwendungen auf dem Gebiet der Quantentechnologie.