Many body localization
Spatial disorder has a tendency to localize quantum mechanical wavefunctions. In the case of noninteracting systems, it is known that generic disorder is always localizing in 1D and 2D, while a minimal disorder strength is required in 3D for localization. .
The fate of localization in the presence of interactions is the subject of Many Body localization.
Starting from delocalized limit, effects of interactions have been studied perturbatively starting with 1980's (Altruler, Aronov, Khmelnitksy; Finkelstein; Fukuyama; ...); however, being perturbative in nature, this approach cannot give a reliable answer whether localization survives any finite interactions.
An alternative approach was to start from the strongly localized regime (Mott; Efros, Shklovskii; ...), and include kinetic energy (quantum tunneling) perturbatively. It appears that localization in this strongly disordered limit can be stable.
In the late 1990's, an experimental discovery of apparently delocalized metallic phases in 2D electron gases (Savchenko, ...) has prompted renewed theoretical and experimental interest in the problem. In early 2000's  a non-perturbative resummation of interactions has been proposed, which precipitated renewed theoretical efforts. The new generation utilized concepts from quantum information theory (entangement entropy) and quantum thermodynamics (Eigenstate Thermalization Hypothesis), combined with a wide range of numerical techniques (exact diagonalization, DMRG, ...).
The results so far include
- discovery/verification of prototypical many-body localized (MBL) phases
- better understanding of MBL-thermal transition
- proposals for many body time crystals.
Many fundamental questions, however, remain, including
- does MBL exits in dimensions higher than 1D?
- can rare fluctuations destroy MBL, even if on extremely long time scales?
- P. W. Anderson, “Absence of diffusion in certain random lattices,” Phys. Rev. 109, 1492–1505 (1958).
- D. M. Basko, I. L. Aleiner, and B. L. Altshuler, “Metal?insulator transition in a weakly interacting manyelectron system with localized single-particle states,” Annals of Physics 321, 1126–1205 (2006).