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

TL;DR

This paper introduces a new computational method based on Localisation Landscape Theory to accurately determine the eigenstate localisation length at very low energies in disordered systems, overcoming limitations of traditional approaches.

Contribution

The paper develops a novel LLT-based approach for calculating low-energy localisation lengths, validated against exact diagonalisation, and discusses its applicability and limitations at higher energies.

Findings

Effective potential from LLT closely matches physical potential at low energies.

The method accurately computes localisation length for maximally localised eigenstates.

Breakdown of the method at higher energies where tunnelling picture fails.

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 confirm that the Hamiltonian with the effective potential of LLT has very similar low energy eigenstates to that with the physical potential, justifying the crucial role the effective potential plays in our new method. We proceed to use LLT to calculate the localisation length for very low-energy, maximally localised eigenstates, as defined by the length-scale of exponential decay of the eigenstates, (manually) testing our findings against exact diagonalisation. We then describe several mechanisms by which the eigenstates spread out at higher energies where the tunnelling-in-the-effective-potential picture breaks down, and explicitly demonstrate that our method is no longer applicable in this regime. We place our computational scheme in context by explaining the connection to the more general problem of multidimensional tunnelling and discussing the approximations involved. Our method of calculating the localisation length can be applied to (nearly) arbitrary disordered, continuous potentials at very low energies.