Higher-order localization landscape theory of Anderson localization

TL;DR

This paper introduces a higher-order landscape theory for Anderson localization, providing a recursive method to estimate eigenenergies and wave functions in disordered quantum systems, with applications demonstrated in various 1D and 2D models.

Contribution

The paper develops a novel higher-order landscape framework that improves localization analysis by iteratively refining eigenenergy and wave function estimates in disordered Hamiltonians.

Findings

Accurate eigenenergy estimations for various disordered potentials.

Wave function approximations closely match true localized states.

Method applicable to both 1D and 2D disordered systems.

Abstract

For a Hamiltonian ${\hat H}$ containing a position-dependent (disordered) potential, we introduce a sequence of landscape functions $u_n(\vec{r})$ obeying ${\hat H} u_n(\vec{r}) = u_{n-1}(\vec{r})$ with $u_0(\vec{r}) = 1$. For $n \to \infty$, $1/v_n(\vec{r}) = u_{n-1}(\vec{r})/u_{n}(\vec{r})$ converges to the lowest eigenenergy $E_1$ of ${\hat H}$ whereas $u_{\infty}(\vec{r})$ yields the corresponding wave function $\psi_1(\vec{r})$. For large but finite $n$, $v_n(\vec{r})$ can be approximated by a piecewise constant function $v_n(\vec{r}) \simeq v_n^{(m)}$ for $\vec{r} \in \Omega_m$ and yields progressively improving estimations of eigenenergies $E_m = 1/v_n^{(m)}$ of locally fundamental eigenstates $\psi_m(\vec{r}) \propto u_{n}(\vec{r})$ in spatial domains $\Omega_m$. These general results are illustrated by a number of examples in one dimension: box potential, sequence of randomly placed infinite potential barriers, smooth and spatially uncorrelated random potentials, quasiperiodic potential, as well as for the uncorrelated random potential in two dimensions.