Laboratoire de Physique Théorique
Toulouse - UMR 5152

In the first part of the talk, I will discuss a generalization of the localization landscape we recently introduced to study highly excited states in Anderson localization models. After a brief introduction of the original localization landscape that gives direct access to the localization properties of bottom-of-band eigenstates in non-interacting disordered systems without having to diagonalize the Hamiltonian, I will introduce our modified $L^2$ landscape, show its accuracy in a variety of archetypal models of Anderson localization in one and two dimensions and compare our approach to other landscape methods, bringing new insights on their strengths and limitations. In the second part of the talk, I will slightly change focus and present some of our works on many-body localization. In the presence of strong disorder, interacting systems can localize and avoid thermalization due to the emergence of an extensive set of local integrals of motions. I will focus on the entanglement structure of the wave-functions at the phase transition between ergodic and MBL phases. After an overview of the properties of the many-body localization, I will discuss how we can access the entanglement structure of the wave-function. Critical states close to the transition have a structure compatible with fractal or multiscale-entangled states, characterized by entanglement at multiple levels : small strongly entangled clusters are weakly entangled together to form larger clusters. The critical point therefore features subthermal entanglement and a power-law distributed cluster size, while the localized phase presents an exponentially decreasing cluster distribution. These results are consistent with some of the recently proposed phenomenological renormalization-group schemes characterizing the many-body localized critical point, and may serve to constrain other such schemes.