Quantum systems of many particles, which satisfy the ergodicity hypothesis, are conventionally described by statistical mechanics. However, not all quantum systems are ergodic, with many-body localization providing a generic mechanism of ergodicity breaking by disorder. Many-body localized (MBL) systems remain perfect insulators at non-zero temperature, which do not thermalize and therefore cannot be described using statistical mechanics. In the review “Ergodicity, Entanglement and Many-Body Localization” arXiv:1804.11065 written together with D. Abanin, E. Altman, and I. Bloch, we summarize recent theoretical and experimental advances in studies of MBL systems, focusing on the new perspective provided by entanglement and non-equilibrium experimental probes such as quantum quenches.
In order to determine generic non-equilibrium dynamics of the generic local observable O one needs two ingredients. The spectrum of the many-body Hamiltonian and the matrix elements of the operator O in the basis of eigenstates completely specify evolution of the local observable O. Moreover, the matrix elements at small energy differences correspond to the long-time dynamics that may be very hard to access in the thermalizing system.
Using these insights, we studied the properties of the matrix elements across the many-body localization transition in the recent preprint arXiv:1610.02389. In particular, we used matrix elements to define the (many-body) Thouless energy ETh which corresponds to the inverse time scale of relaxation. The behavior of this Thouless energy reveals the critical fan preceding localization transition. In addition, we associate the critical region with the onset of broad distributions and typicality breakdown.