This book presents a complete study of natural orbitals in quantum impurity problems, revealing a certain simplicity in these interacting many-body problems. These systems consist of a few localized degrees of freedom that undergo strong interactions and hybridize with a larger system of free particles; they are central in the study of strongly correlated systems. In a first step, the standard non-perturbative numerical renormalization group method is employed to demonstrate the hierarchical structure of correlations unveiled by natural orbitals. This simplification brought new insights for simulating quantum impurity problems, and a new algorithm is developed to generate an optimized subset of natural orbitals independently of existing methods, going beyond their usual limitations. This algorithm is presented in detail in the book, and a careful benchmark on known results is carried out to guarantee the validity of the method. It is then used to study spatialentanglement structures under various conditions that were not accessible with previous methods, such as representing the electron bath by a realistic 2D square lattice or taking account of static disorder in the metallic host. In the last chapter, the non-interacting problem in the presence of disorder is studied through random matrix theory, reproducing some of the results presented in the previous chapters. The main original result of this chapter lies in the analytical calculation of the joint distribution of one-particle orbitals energies and amplitudes of the impurity, which makes it possible to calculate any disordered averaged local correlation functions. Starting from this result, calculations in the large-N limit are compared with numerical simulations.

Autorentext

Maxime Debertolis received his PhD in theoretical Condensed Matter physics in 2022 under the supervision of S. Florens at Néel Institute (CNRS) in Grenoble, France. His PhD research was focused on the study of quantum impurity problems, which are central to the field of condensed matter. During his thesis, he proposed a new method to simulate these problems at equilibrium, which has opened up new perspectives for future work in this field. He is currently working as a postdoctoral researcher in the group of D. Luitz at Bonn University, Germany, on theoretical quantum many-problems and quantum simulation, linking problems of condensed matter to the current development of digital quantum computers.