It is characterized by the system failing to reach thermal equilibrium, and retaining a memory of its initial condition in local observables for infinite times.
However, a quantum system generically contains a macroscopic number of degrees of freedom, but can only be probed through few-body measurements which are local in real space.
This question can be formalized by considering the quantum mechanical density matrix ρ of the system.
If the system is divided into a subregion A (the region being probed) and its complement B (everything else), then all information that can be extracted by measurements made on A alone is encoded in the reduced density matrix
[3][4][5] Thermalizing systems therefore generically have extensive or "volume law" entanglement entropy at any non-zero temperature.
fails to approach a thermal density matrix even in the long time limit, and remains instead close to its initial condition
In 1980 Fleishman and Anderson[23] demonstrated the phenomenon survived the addition of interactions to lowest order in perturbation theory.
In a 1998 study,[24] the analysis was extended to all orders in perturbation theory, in a zero-dimensional system, and the MBL phenomenon was shown to survive.
A series of numerical works[27][14][28] [29] provided further evidence for the phenomenon in one dimensional systems, at all energy densities (“infinite temperature”).
It is now believed that MBL can arise also in periodically driven "Floquet" systems where energy is conserved only modulo the drive frequency.
It was conjectured[34][35] (and proven by Imbrie) that a similar extensive set of local integrals of motion should also exist in the MBL phase.
If the original Hamiltonian is perturbed, the l-bits get redefined, but the integrable structure survives.
[36] A form of localization-protected quantum order, arising only in periodically driven systems, is the Floquet time crystal.
[42][43][44][45] Most of these experiments involve synthetic quantum systems, such as assemblies of ultracold atoms or trapped ions.