Abstract

In 1958 P. W. Anderson showed that eigenstates of a single-particle quantum Hamiltonian in the presence of disorder can be localized in space. In the same paper he had speculated on the possibility of lack of thermalization in an isolated quantum system even in the presence of interactions. There is now growing evidence that such an isolated system in the presence of strong disorder fails to \it{equilibrate}. This phenomena is being referred to as \it{many-body localization} (MBL). I will introduce the various defining characteristics of the MBL phase and the measures which can be used to distinguish it from the ergodic phase. Based on these I will show numerical evidence of the hypothesized phase-transition between the MBL and thermal phases in a short-ranged model. I will also describe the many-body localization-delocalization (MBLD) transition in the quantum random energy model. It is the “simplest” mean-field model for the equilibrium spin-glass transition. Its analytical tractability opens the possibility to develop a mean-field understanding of the dynamical critical point. Due to the violation of ergodicity MBL even allows the existence of symmetry-breaking and topological order even in highly excited eigenstates, which would normally be destroyed by thermal fluctuations at equilibrium. I will present a phenomenological description of this localization protected quantum order.