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

Motivated by experimental progress in cold atomic systems, we use and advance Localisation Landscape Theory (LLT), to examine two-dimensional systems with point-like random scatterers. 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 confirm that the Hamiltonian with the effective potential of LLT has very similar low energy eigenstates to that with the physical potential. Next, we use LLT to compute the localisation length for very low-energy, maximally localised eigenstates and test our method against exact diagonalisation. Furthermore, we propose a transmission experiment that optimally detects Anderson localisation, and demonstrate how one may extract a length scale which is correlated with (and in general smaller than) the localisation length. 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, to find that these can be useful in detecting and quantifying Anderson localisation in a transmission experiment, with the only disadvantage being the inability to resolve the energy dependence of the localisation length. 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.