Thesis presented November 07, 2024

Abstract:
The physics of N-body quantum systems reveals the emergence of numerous unexpected collective phenomena, such as the 0.7 anomaly, the Kondo effect, or high critical temperature superconductivity. From a theoretical standpoint, describing these systems is complex due to the exponential increase in the size of the state space as the number of particles increases. For example, 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, paved the way for many advances. In this thesis, we focus on the physics of quantum dots, described by the Anderson model. This model presents rich phenomenology, both in the charge domain (Coulomb blockade) and in the spin domain (Kondo effect). While this model is well understood in equilibrium, we propose here an innovative technique to study its properties in the non-equilibrium regime, a historically challenging area to address. Our approach allows for exact calculations with controlled errors. This method is an evolution of the "Diagrammatic Monte Carlo" technique, which systematically computes a very large number of Feynman diagrams. Traditionally, this technique stochastically samples the diagram space. In our work, we replaced this approach with a machine learning method, based on tensor networks and the cross-interpolation algorithm. The latter detects and exploits the specific structure of the diagram space, significantly speeding up the calculations. Its major advantage is that it is insensitive to the sign problem that limits stochastic approaches. Thanks to this new technique, we were able to calculate all the diagrams up to much higher orders than before (<30), which allowed us to unveil the physics of non-equilibrium quantum dots with unparalleled precision.

Keywords:
quantum many-body problem,tensor network, condensed matter, Anderson model, tensor cross-interpolation, quantum impurity