Dual landscapes in Anderson localization on discrete lattices

In Anderson localization, electronic states are exponentially localized despite the absence of classical confinement, this localization being
explained as originating from the destructive interference of waves reflected in the random atomic potential. Despite numerous theoretical advances, such as the prediction by the scaling theory of the lower critical dimension of the Anderson transition, there was until recently no general formalism capable to accurately pinpoint the spatial location of these localized modes for any given potential, nor to predict the exact energy at which delocalized modes would begin to form.

Recently, a new theory has been proposed, unveiling in continuous media a direct relationship between any specific realization of the random potential and the corresponding location of localized states. It has been demonstrated that the boundaries of the localization regions,
which cannot be deduced by directly looking at the bare random potential, can be accurately retrieved as the valleys lines of a ``hidden landscape" $u(x)$ which is the solution of a Dirichlet problem with uniform right-hand side for the same Hamiltonian.

In this talk, we show that not only the exact same theory can be extended to the case of a tight-binding Hamiltonian defined on a discrete lattice, but also that, contrary to the continuous case, two different types of localization occur here. First, localization of low energy states can be predicted using a discrete analog of the landscape $u(x)$ defined in the continuous situation. Secondly, the discreteness of the system also triggers a strong localization of states of typical wavelength of the order of the lattice spacing (corresponding to the top of the band for a periodic potential). We show that this localization can also be studied in the framework of the landscape theory, with a different operator than the original Hamiltonian and, respectively, a different landscape.