${\mathcal{L}}^2$ localization landscape for highly-excited states

The localization landscape gives direct access to the localization of bottom-of-band eigenstates in non-interacting disordered systems. We generalize this approach to eigenstates at arbitrary energies in systems with or without internal degrees of freedom by introducing a modified ${\mathcal{L}}^2$-landscape, and we demonstrate its accuracy in a variety of archetypal models of Anderson localization in one and two dimensions. This ${\mathcal{L}}^2$-landscape function can be efficiently computed using hierarchical methods that allow evaluating the diagonal of a well-chosen Green function. We compare our approach to other landscape methods, bringing new insights on their strengths and limitations. Our approach is general and can in principle be applied to both studies of topological Anderson transitions and many-body localization.