Od kvantnega kaosa do zloma ergodičnosti v izoliranih mnogodelčnih kvantnih sistemih

The ETH represents a breakthrough in many-body physics since it links thermalization of physical observables with the applicability of RMT. This framework is widely believed to hold for an overwhelming majority of physical systems, though exceptions, where conventional ETH is violated, include integrability, single-particle chaos, many-body localization, many-body scars, and Hilbert-space fragmentation. However, the mechanism of the ETH breakdown remains elusive. In this thesis, we introduce a novel scenario in many-body quantum systems, dubbed ${it fading ergodicity}$ regime, which establishes a link between the conventional ETH and non-ergodic behavior. This framework predicts an exponent, $eta$, governing eigenstate fluctuations in the matrix elements of local observables. As a consequence, slow dynamics emerge as the perturbations become vanishingly small, and the Thouless energy matches the Heisenberg scale at the critical point. We conjecture this scenario to be relevant for the description of many-body systems, where the divergent relaxation time is described by the Fermi golden rule. We provide numerical and analytical arguments for its validity in the quantum sun model and related RMT models. We continue our discussion by exploring the universality of the fading ergodicity scenario. Remarkably, we show that fading ergodicity leads to a maximally divergent peak in the fidelity susceptibility, establishing the onset of ${it maximal chaos}$ at the critical point. Furthermore, we extend this framework to finite energy density, with both fading ergodicity and maximal chaos following the many-body mobility edge to high precision. Finally, we present a scaling theory of many-body ergodicity breaking, which exhibits a critical exponent $nu = 1$ at the critical point, characterizing the universality class of fading ergodicity. However, in contrast to the well established Anderson localization transition, we argue that the one-parameter scaling theory is insufficient.