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This paper originates from lectures delivered at the summer school"Fundamental Problems in Statistical Physics XV" in Bruneck, Italy, in 2021. Wegive a brief and limited introduction into ergodicity-breaking induced bydisorder. As the title suggests, we include a simple yet rigorous and originalresult: For a strongly disordered quantum chain, we exhibit a full set ofquasi-local quantities whose dynamics is negligible up to times of order$\exp\{ c (\log W)^{2-\epsilon}\}$, with $\epsilon < 1$, $c$ a numericalconstant, and $W$ the disorder strength. Such a result, that is often referredto as "quasi-localization", can in principle be obtained in other systems aswell, but for a disordered quantum chain, its proof is relatively short andtransparent.

Rigorous and simple results on very slow thermalization, or quasi-localization, of the disordered quantum chain