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

Abstract

Motivated by rapid experimental progress in ultra-cold atomic systems, we aim to provide a simple, intuitive description of Anderson localisation that allows for a direct quantitative comparison to experimental data, as well as yielding novel insights. To this end, we advance, employ and validate a recently-discovered theory -Localisation Landscape Theory (LLT) -which has unparalleled strengths and advantages, both computational and conceptual, over alternative methods. We focus on two-dimensional systems with point-like random scatterers, although an analogous study in other dimensions and with other types of disorder would proceed similarly. 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 Schrödinger 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.

Computing the eigenstate localisation length at very low energies from Localisation Landscape Theory

“…Precisely the same trend is seen in Fig. 8 of [52], where the approximate eigenstates and eigenvalues are reconstructed from the localisation landscape u, avoiding numerical diagonalisation. This is very likely linked to the fact that in both cases, the approximate LLT eigenstates are a little more spread out than the exact.…”

“…This system could be experimentally realised with cold atoms as in [14]. A more detailed description of the system can be found in [52].…”

“…We therefore begin our investigation by directly diagonalising the Hamiltonian and inspecting the eigenstates and energies, with the goals of (a) gaining intuition for our system and (b) checking whether useful quantitative predictions may be readily obtained in this framework. Details on the numerical implementation are given in appendix A of [52].…”

“…For the rest of the article, familiarity with LLT is certainly helpful. This background knowledge can be obtained by reading the original papers [44][45][46][47][48][49][50], or a succinct review of the key facts found in [52] (we use the same notation here as in [52]). Note that the present article is one out of five related publications, available as a single, coherent report [52].…”

“…Conclusions are presented in section 7 and several ideas are discussed as directions for a possible forthcoming investigation. Details about our computational techniques and the numerical methods employed can be found in the appendices of [52].…”