# Localisation and transport in bidimensional random models with separable Hamiltonians

## TL;DR

This paper investigates two 2D random models with separable Hamiltonians, analyzing how disorder affects localization in angular or radial directions, and compares these behaviors to 1D models, highlighting design and finite-size effects.

## Contribution

It introduces and analyzes two 2D models with separable Hamiltonians, exploring localization phenomena and their relation to 1D counterparts, with implications for wavefunction control.

## Key findings

- Disorder localizes either angular or radial eigenfunctions.
- Correlated disorder can tailor localization length and directionality.
- Finite-size and resonance effects influence eigenfunction structure.

## Abstract

We consider two bidimensional random models characterised by the following features: a) their Hamiltonians are separable in polar coordinates and b) the random part of the potential depends either on the angular coordinate or on the radial one, but not on both. The disorder correspondingly localises the angular or the radial part of the eigenfunctions. We analyse the analogies and the differences which exist between the selected 2D models and their 1D counterparts. We show how the analogies allow one to use correlated disorder to design a localisation length with pre-defined energy dependence and to produce directional localisation of the wavefunctions in models with angular disorder. We also discuss the importance of finite-size and resonance effects in shaping the eigenfunctions of the model with angular disorder; for the model with disorder associated to the radial variable we show under what conditions the localisation length coincides with the expression valid in the 1D case.