Thermalizing quantum systems are conventionally described by statistical
mechanics at equilibrium. However, not all systems fall into this category,
with many body localization providing a generic mechanism for thermalization to
fail in strongly disordered systems. 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 this Colloquium
we review 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. Theoretically,
MBL systems exhibit a new kind of robust integrability: an extensive set of
quasi-local integrals of motion emerges, which provides an intuitive
explanation of the breakdown of thermalization. A description based on
quasi-local integrals of motion is used to predict dynamical properties of MBL
systems, such as the spreading of quantum entanglement, the behavior of local
observables, and the response to external dissipative processes. Furthermore,
MBL systems can exhibit eigenstate transitions and quantum orders forbidden in
thermodynamic equilibrium. We outline the current theoretical understanding of
the quantum-to-classical transition between many-body localized and ergodic
phases, and anomalous transport in the vicinity of that transition.
Experimentally, synthetic quantum systems, which are well-isolated from an
external thermal reservoir, provide natural platforms for realizing the MBL
phase. We review recent experiments with ultracold atoms, trapped ions,
superconducting qubits, and quantum materials, in which different signatures of
many-body localization have been observed. We conclude by listing outstanding
challenges and promising future research directions.Comment: (v2) minor changes, added one figure and expanded bibliography; (v1)
colloquium-style review on many-body localization; 29 pages, 11 figures;
comments are welcom