Anderson localisation in two dimensions: insights from Localisation Landscape Theory, exact diagonalisation, and time-dependent simulations

Abstract

Motivated by experimental progress in cold atomic systems, we use and further develop Localisation Landscape Theory (LLT), which has unparalleled strengths over alternative methods. We focus on two-dimensional systems with point-like random scatterers, but our methods are neither model nor dimension specific. We begin by showing that exact eigenstates cannot be efficiently used to extract the localisation length. We then provide a comprehensive review of known LLT, and show that the effective potential of LLT can, to some degree, replace the real potential in the Hamiltonian. Next, we use LLT to compute the localisation length and test our method against exact diagonalisation. Furthermore, we propose a transmission experiment that optimally detects Anderson localisation and link the simulated observations of such an experiment to the predictions of LLT. In addition, we study the dimensional crossover from one to two dimensions, providing a new explanation to the established trends. The prediction of a mobility edge coming from LLT is tested by direct Schr\"odinger time evolution and is found to be unphysical. Moreover, we investigate expanding wavepackets, and find interesting differences between wavepackets that are initiated within and outside the disorder. We explain these differences using LLT combined with multidimensional tunnelling. Then, we utilise LLT to uncover a connection between the Anderson model for discrete disordered lattices and continuous two-dimensional disordered systems, which provides powerful new insights. From here, we demonstrate that localisation can be distinguished from other effects by a comparison to dynamics in an ordered potential with all other properties unchanged. Finally, we thoroughly investigate the effect of acceleration and repulsive interparticle interactions, as relevant for current experiments.