Localization and non equilibration in quantum disordered systems, from 1958 to yesterday

Quantum disordered many-body systems represent the most spectacular known failure of statistical mechanics. In such systems, quantum interference effects lead to a breakdown of the ergodic hypothesis. This phenomenon, known as “many-body localization” (MBL), is associated with many other peculiar features, like absence of diffusion and transport, as well as the presence of an extensive set of local conserved quantities. In this talk, after a brief introduction about equilibration and thermalization in closed quantum systems, I will review the most important properties of localized systems, both at the single particle and many-body level. I will also briefly discuss some current topics of research, like the possibility of having MBL in translation invariant systems, and the possibility that non-perturbative effects may reinstate (very slow) equilibration in the thermodynamic limit.