Anderson’s seminal work of 1958 showed how the presence of disorder in a lattice can cause the single-particle eigenstates of a quantum system to become spatially localised. This can be understood in terms of the mismatch in energy levels at different lattice sites, which prevents single-site eigenstates from being hybridised with each other, thus hampering the transport of particles. Since its discovery, Anderson Localisation has been the subject of a vast body of work, and it is now recognized as “one of the fundamental concepts in the physics of condensed matter and disordered systems” [50 Years of Anderson Localization, 2008].

Many-Body localisation

Can Anderson-like localisation occur in interacting many-body quantum systems? This question has been the focus of great theoretical interest in recent years, leading to the definition of a many-body localised (MBL) phase. This phenomenon would provide a mechanism for many-body quantum systems not to thermalise. Characterising the many-body localised phase and identifying the MBL phase transition are important open problems.

Localisation on graphs

Motivated by the study of MBL, there has been renewed interest in the localisation properties of quantum particles on hierarchical lattices, such as the random regular graph. These systems exhibit extended non-ergodic phases, with separate Anderson-like and ergodic transitions. The investigation of fractal properties of such non-ergodic phases could help shed light on their physical properties.

Dynamical Mean Field Theory

In strongly correlated quantum systems, where interactions can not be approximated by a static mean field theory, the nature of localisation can be a contentious issue. Dynamical Mean Field Theory (DMFT) is the leading method used to treat such systems numerically and recent research using DMFT has shed some light of the role of localisation.