A stochastic method to compute the $L^2$ localisation landscape

TL;DR

This paper introduces a stochastic approach for computing the $L^2$ localisation landscape, allowing efficient analysis of localization phenomena in quantum models using sparse matrices and energy filtering techniques.

Contribution

It presents a novel stochastic method for calculating the $L^2$ localisation landscape and incorporates energy filtering to target specific eigenstates, enhancing analysis of localization in quantum systems.

Findings

Effective computation of $L^2$ landscapes using sparse matrix methods.

Application to Anderson localization in 1D and 2D models.

Analysis of localization in quantum Hall effect models.

Abstract

The $L^2$ localisation landscape of L. Herviou and J. H. Bardarson is a generalisation of the localisation landscape of M. Filoche and S. Mayboroda. We propose a stochastic method to compute the $L^2$ localisation landscape that enables the calculation of landscapes using sparse matrix methods. We also propose an energy filtering of the $L^2$ landscape which can be used to focus on eigenstates with energies in any chosen range of the energy spectrum. We demonstrate the utility of these suggestions by applying the $L^2$ landscape to Anderson's model of localisation in one and two dimensions, and also to localisation in a model of the quantum Hall effect.