The physics of N-body quantum systems reveals the emergence of numerous unexpected collective phenomena, such as the 0.7 anomaly, the Kondo effect, and high critical temperature superconductivity. From a theoretical perspective, describing these systems is challenging due to the exponential growth of the state space as the number of particles increases. For instance, the quantitative understanding of the Kondo effect required the development of new concepts and mathematical tools, including the (numerical) renormalization group. This pioneering work, which earned Wilson the Nobel Prize in 1978, laid the foundation for many subsequent advances.In this thesis, we focus on the physics of quantum dots, described by the Anderson model. This model exhibits rich phenomenology, both in the charge domain (Coulomb blockade) and the spin domain (Kondo effect). While this model is well understood in equilibrium, we propose an innovative technique to study its properties in the non-equilibrium regime—a historically challenging area to address. Our approach enables exact calculations with controlled errors.This method builds on the "Diagrammatic Monte Carlo" technique, which systematically computes a large number of Feynman diagrams. Traditionally, this technique stochastically samples the space of diagrams. In our work, we replaced this approach with a machine learning method based on tensor networks and the cross-interpolation algorithm. The latter identifies and leverages the specific structure of the diagram space, greatly accelerating the calculations. A key advantage of this method is its insensitivity to the sign problem that limits stochastic approaches.Using this new technique, we were able to compute diagrams up to much higher orders than previously possible (<30), enabling us to explore the physics of non-equilibrium quantum dots with unprecedented precision.
The physics of N-body quantum systems reveals the emergence of numerous unexpected collective phenomena, such as the 0.7 anomaly, the Kondo effect, and high critical temperature superconductivity. From a theoretical perspective, describing these systems is challenging due to the exponential growth of the state space as the number of particles increases. For instance, the quantitative understanding of the Kondo effect required the development of new concepts and mathematical tools, including the (numerical) renormalization group. This pioneering work, which earned Wilson the Nobel Prize in 1978, laid the foundation for many subsequent advances.In this thesis, we focus on the physics of quantum dots, described by the Anderson model. This model exhibits rich phenomenology, both in the charge domain (Coulomb blockade) and the spin domain (Kondo effect). While this model is well understood in equilibrium, we propose an innovative technique to study its properties in the non-equilibrium regime—a historically challenging area to address. Our approach enables exact calculations with controlled errors.This method builds on the "Diagrammatic Monte Carlo" technique, which systematically computes a large number of Feynman diagrams. Traditionally, this technique stochastically samples the space of diagrams. In our work, we replaced this approach with a machine learning method based on tensor networks and the cross-interpolation algorithm. The latter identifies and leverages the specific structure of the diagram space, greatly accelerating the calculations. A key advantage of this method is its insensitivity to the sign problem that limits stochastic approaches.Using this new technique, we were able to compute diagrams up to much higher orders than previously possible (<30), enabling us to explore the physics of non-equilibrium quantum dots with unprecedented precision.
Calcul numérique exact des diagrammes de Feynman pour la physique de Kondo hors équilibre en utilisant l'Interpolation de Tenseur par produit de Croix La physique des systèmes quantiques à N-corps révèle l’émergence de nombreux phénomènes collectifs inattendus, tels que l'anomalie 0.7, l'effet Kondo ou la supraconductivité à haute température critique. D'un point de vue théorique, la description de ces systèmes est complexe en raison de l’augmentation exponentielle de la taille de l’espace des états en fonction du nombre de particules. Par exemple, la compréhension quantitative de l’effet Kondo a nécessité le développement de nouveaux concepts et outils mathématiques, dont le groupe de renormalisation (numérique). Ce travail pionnier, qui valut à Wilson le prix Nobel en 1978, a ouvert la voie à de nombreuses avancées.Dans cette thèse, nous nous intéressons à la physique des boîtes quantiques, décrites par le modèle d’Anderson. Ce modèle présente une riche phénoménologie, tant dans le domaine de la charge (blocage de Coulomb) que dans celui du spin (effet Kondo). Si ce modèle est bien compris en régime d'équilibre, nous proposons ici une technique novatrice pour étudier ses propriétés dans le régime hors équilibre, un domaine historiquement difficile à traiter. Notre approche permet un calcul exact avec des erreurs contrôlées.Cette méthode est une évolution du « Monte Carlo diagrammatique », qui permet de calculer de manière systématique un très grand nombre de diagrammes de Feynman. Traditionnellement, cette technique échantillonne l’espace des diagrammes de façon stochastique. Dans notre travail, nous avons remplacé cette approche par une méthode d’apprentissage machine, basée sur les réseaux de tenseurs et l'algorithme d'interpolation par produit de croix. Ce dernier détecte et exploite la structure spécifique de l’espace des diagrammes, accélérant ainsi considérablement les calculs. Il présente l’avantage majeur d’être insensible au problème de signe qui limite les approches stochastiques.Grâce à cette nouvelle technique, nous avons pu calculer l’ensemble des diagrammes jusqu’à des ordres bien plus élevés qu’auparavant (<30), ce qui nous a permis de dévoiler la physique des boîtes quantiques hors équilibre avec une précision inégalée.
We use tensor network techniques to obtain high-order perturbative diagrammatic expansions for the quantum many-body problem at very high precision.The approach is based on a tensor train parsimonious representation of the sum of all Feynman diagrams, obtained in a controlled and accurate way with the tensor cross interpolation algorithm.It yields the full time evolution of physical quantities in the presence of any arbitrary time-dependent interaction.Our benchmarks on the Anderson quantum impurity problem, within the real-time nonequilibrium Schwinger-Keldysh formalism, demonstrate that this technique supersedes diagrammatic quantum Monte Carlo by orders of magnitude in precision and speed, with convergence rates 1=N 2 or faster, where N is the number of function evaluations.The method also works in parameter regimes characterized by strongly oscillatory integrals in high dimension, which suffer from a catastrophic sign problem in quantum Monte Carlo calculations.Finally, we also present two exploratory studies showing that the technique generalizes to more complex situations: a double quantum dot and a single impurity embedded in a two-dimensional lattice.
In this thesis, we describe methods for solving quantum impurity models out of equilibrium, and consider several applications of these methods. Quantum impurity models describe systems in which a small subsystem (the impurity) is coupled to a large external environment (the bath). These models are useful in describing a wide variety of physical systems, including quantum dots, absorbed adatoms, and molecular junctions. They are also important for the study of lattice models, such as the Hubbard model, where they appear as auxiliary problems within the framework of dynamical mean field theory (DMFT). Many interesting phenomena in these systems, such as transport through a quantum dot, or optical pumping of a Mott insulator, occur away from equilibrium. Theoretical description of these phenomena requires methods that are capable of handling both real time evolution and strong correlation effects presenting a formidable challenge. In this thesis, we describe several methods for the solution of non-equilibrium quantum impurity models which meet this challenge. These methods are based on an expansion in the coupling between the impurity and the bath known as the hybridization expansion. Chapter 1 gives an introduction to the problems addressed in this thesis. Chapter 2 provides a brief overview of Keldysh formalism for nonequilibrium problems. Chapter 3 describes the bare hybridization expansion for the Anderson impurity model as well as its bold extensions. In particular we derive the non-crossing approximation (NCA), the one-crossing approximation (OCA), the inchworm expansion, and a bold expansion for the impurity Green's function. Chapter 4 describes the implementation of a quantum Monte Carlo algorithm based on the inchworm expansion. The rest of the thesis describes applications of these algorithms to problems of physical interest. Chapters 5 and 6 apply the methods previously described to understanding the Kondo effect in quantum dot systems. The first chapter examines the behavior of the Kondo effect under an applied voltage bias, a classic non-equilibrium strongly-correlated problem. The second chapter examines different ways of measuring the cloud of electrons which screen the impurity spin in the Kondo regime. Chapters 7 and 8 apply the same methods to the Hubbard model using the Dynamical mean-field theory (DMFT) mapping. The first chapter studies the Hubbard model in equilibrium using the numerically exact inchworm quantum Monte Carlo method. The second chapter studies the possibility of dynamic switching between the metal and Mott-insulator states of the Hubbard model using the one-crossing approximation (OCA). Finally chapter 9 gives some overall conclusions and suggests future directions for the development of hybridization expansion methods.
This thesis contributes to the field of transport through quantum dots. These devices allow for a controlled study of quantum transport and fundamental physical effects, like the Kondo effect. In this thesis we will focus on dots that are well described by generalized Anderson impurity models, where the discrete levels of the quantum dot are tunnel-coupled to fermionic reservoirs. The model parameters, like level energy and width, can be tuned in experiments. Therefore these systems constitute a valuable arena for testing experiment against theory and vice versa. In order to describe these strongly correlated systems, we employ the numerical renormalization group method. This allows us to address both longstanding questions concerning experimental results and new physical phenomena in these fundamental models. This thesis consists of three major projects. The first and most extensive one is concerned with the phase of the transmission amplitude through a quantum dot. Measurements of many-electron quantum dots with small level spacing reveal universal phase behaviour, a result not fully understood for almost 10 years. Recent experiments have seen that, contrarily, for dots with only a few electrons, i.e. large level spacing, the phase depends on the mesoscopic dot parameters. Analyzing a multi-level Anderson model, we show that the generic feature of the two regimes can be reproduced in the regime of overlapping levels or well separated levels, respectively. Thereby the universal character follows from Fano-type antiresonances of the renormalized single-particle levels. Moderate temperature supports the universal character. In the mesoscopic regime, we also investigate the effect of Kondo correlations on the transmission phase. In a second project we analyze a quantum dot coupled to a superconducting reservoir. In contrast to previous belief, the energy resolution of our method is not restricted by the energy scale of the superconducting gap, leading to new insights into the method. The high resolution allows us to resolve sharp peaks in the spectral function that emerge for a certain regime of parameters. A third project deals with a quantum dot coupled to two independent channels, a system known to exhibit non-Fermi liquid behaviour. We investigate the existence of the non-Fermi liquid regime when driving the system out of the Kondo regime by emptying the dot. We find that the extent of the non-Fermi liquid regime strongly depends on the mechanisms that couple impurity and reservoirs but prevent mixing of the latter.
In this thesis I investigate several aspects of local quantum criticality, a concept of key importance in a number of physical contexts ranging from critical heavy fermion compounds to quantum dot systems. Quantum critical points are associated with second order phase transitions at zero temperature. In contrast to their finite-temperature counterparts, the zero-point motion cannot be neglected near a quantum critical point. As a result, the incorporation of quantum dynamics leads to an effective dimension larger than the spatial dimension of the system for the order parameter fluctuations within the Ginzburg-Landau-Wilson treatment of criticality. This so-called quantum-to-classical mapping works well for the critical properties in insulating systems but apparently fails in systems containing gapless fermions. This has been experimentally most clearly been demonstrated within a particular class of intermetallic compounds called heavy fermions. A particular way in which the Ginzburg-Landau-Wilson paradigm fails is for critical Kondo destruction that seems to underlie the unconventional quantum criticality seen in the heavy fermions. I focus on studying the properties of critical Kondo destruction and the emergence of energy-over-temperature-scaling in systems without spatial degrees of freedom, i.e., so-called quantum impurity systems. In particular, I employ large-N techniques to address critical properties of this class of quantum phase transitions in and out of equilibrium. As quantum critical systems are characterized by a scale-invariant spectrum with many low-lying excitations, it may appear that any perturbation can lead to a response beyond the linear response regime. Understanding what governs the non-linear response regime near quantum criticality is an interesting area. Here, I first present a path integral version of the Schrieffer-Wolff transformation which relates the functional integral form of the partition function of the Anderson model to that of its effective low-energy model. The equivalence between the low-energy sector of the Anderson model in the Kondo regime and the spin-isotropic Kondo model is usually established via a canonical transformation performed on the Hamiltonian, followed by a projection. The resulting functional integral assumes the form of a spin path integral and includes a geometric phase factor, i.e. a Berry phase. The approach stresses the underlying symmetries of the model and allows for a straightforward generalization of the transformation to more involved models. As an example of the efficiency of the approach I apply it to a single electron transistor attached to ferromagnetic leads and derive the effective low-energy model of such a magnetic transistor. As Kondo screening is a local phenomenon, it and its criticality can be studied using the appropriate impurity model. A general impurity model to study critical Kondo destruction is the pseudogap Bose-Fermi Kondo model. Here, I concentrate on the multi-channel version of the model using the dynamical large-N study. This model allows to study the non-trivial interplay between two different mechanisms of critical Kondo destruction. The interplay of two processes that can each by itself lead to critical Kondo destruction. The zero-temperature residual entropy at various fixed points for the model is also discussed. The two channel Anderson model exhibits several continuous quantum phase transitions between weak- and strong-coupling phases. The non-crossing approximation (NCA) is believed to give reliable results for the standard two-channel Anderson model of a magnetic impurity in a metal. I revisit the reliability of the NCA for the standard two channel Anderson model (constant conduction electron density of states) and investigate its reliability for the two-channel pseudogap Anderson model. This is done by comparing finite-temperature, finite-frequency solutions of the NCA equations and asymptotically exact zero-temperature NCA solutions with numerical renormalization-group calculations. The phase diagram of this model is well established. The focus here will be on the dynamical scaling properties obtained within the NCA. Finally, I study the thermal and non-thermal steady state scaling functions and the steady-state dynamics of the pseudogap Kondo model. This model allows us to study the concept of effective temperatures near fully interacting as well as weak-coupling fixed points and compare the out-of-equilibrium scaling properties of critical Kondo destruction to those of the traditional spin-density wave (SDW) scenario. The differences I identify can be experimentally probed. This may be helpful in identifying the nature of the quantum critical points observed in certain heavy fermion compounds.
Kondo physics first surfaced into the scientific world in 1934 via an unexpected observations made by a group of experimentalists, who measured the temperature-dependent resistance of gold wires in their laboratory at the University of Leiden. When decreasing the temperature below 4 Kelvin, they observed an unexpected increase of the resistance, which resulted in a resistance minimum that was not accounted for by the theoretical understanding at that time. Only 30 years later, Jun Kondo, a Japanese physicist from Tokyo, was the first to explain this curiosity, known as the Kondo effect, under the assistance of the Kondo model. Although his findings explained the increased resistance, a more fundamental problem was raised: the Kondo problem, an unphysical divergence of the resistance at very low temperatures. To solve this problem required the development of more advanced methods beyond the scope of pure perturbative methods, namely the use of renormalization group (RG) methods. More than twenty years after Kondo’s first steps towards the solution of the resistance minimum, the Kondo effect experienced a scientific revival due to the emergence of quantum dot systems in solid state physics. A quantum dot is a very localized and spatially restricted area within a metal, where only a few electrons fit in the area. The limited spatial spread constrains the possible positions for the electrons on the dot, and forces them into strong interactions with each other–they are strongly correlated. In quantum dot systems, not only the Kondo effect in equilibrium situations between the dot and its surrounding reservoirs can be studied, but also the effects of nonequilibrium situations, where a finite bias voltage drives a current through the quantum dot. However, the nonequilibrium Kondo effect is far less explored compared to its equilibrium counterpart and it requires the development of new methods beyond established equilibrium methods. To fill this gap, we use the recently developed real-time renormalization group (RTRG) method in the weak-coupling regime and its extension to the crossover regime, from weak coupling to strong coupling. Furthermore, different combinations between the number of electrons on the dot and the number of reservoirs of electrons in the surrounding metal allow for distinct physical situations, which can be studied within the framework of the RTRG method. We subsequently characterize the properties of the the overscreened, the underscreened, and the fully screened Kondo model by analyzing their physical observables and, in parts, their time evolution. The perturbative foundation of the RTRG method gives us the opportunity to gather new analytic insights for higher spin Kondo systems, on the one hand, and provides for a qualitative comparison of obtained results to various other methods and analytic results, on the other hand.