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

Abstract

While Anderson localisation is largely well-understood, its description has traditionally been rather cumbersome. A recently-developed theory -Localisation Landscape Theory (LLT) -has unparalleled strengths and advantages, both computational and conceptual, over alternative methods. To begin with, we demonstrate that the localisation length cannot be conveniently computed starting directly from the exact eigenstates, thus motivating the need for the LLT approach. Then, we reveal the physical significance of the effective potential of LLT, justifying the crucial role it plays in our new method. We proceed to use LLT to calculate the localisation length, as defined by the length-scale of exponential decay of the eigenstates, (manually) testing our findings against exact diagonalisation. We place our computational scheme in context by explaining the connection to the more general problem of multidimensional tunnelling and discussing the approximations involved. The conceptual approach behind our method is not restricted to a specific dimension or noise type and can be readily extended to other systems.