[Paper Review] Effective confining potential of quantum states in disordered media

Douglas N. Arnold, Guy David+3 more|arXiv (Cornell University)|May 11, 2015

Spectral Theory in Mathematical Physics3 citations

TL;DR

This paper introduces the reciprocal of the localization landscape, $ W = 1/u $, as an effective confining potential that governs the spatial confinement and long-range exponential decay of quantum states in disordered media. By transforming the Schrödinger equation via $ \psi = u\varphi $, the authors show that $ W $ acts as a semiclassical potential whose barriers mediate tunneling and explain Anderson localization through multiple barrier penetrations, leading to a highly accurate approximation of the integrated density of states via Weyl's law applied to $ W $.

ABSTRACT

The amplitude of localized quantum states in random or disordered media may exhibit long range exponential decay. We present here a theory that unveils the existence of an effective potential which finely governs the confinement of these states. In this picture, the boundaries of the localization subregions for low energy eigenfunctions correspond to the barriers of this effective potential, and the long range exponential decay characteristic of Anderson localization is explained as the consequence of multiple tunneling in the dense network of barriers created by this effective potential. Finally, we show that the Weyl's formula based on this potential turns out to be a remarkable approximation of the density of states for a large variety of one-dimensional systems, periodic or random.

Motivation & Objective

• To explain the mechanism behind the long-range exponential decay of localized quantum states in disordered systems, despite the absence of an external confining potential.
• To identify a new effective potential $ W = 1/u $ derived from the localization landscape $ u $, which governs the confinement and tunneling behavior of quantum states.
• To demonstrate that Weyl's law applied to $ W $ yields a remarkably accurate approximation of the integrated density of states (IDOS) in one-dimensional systems, outperforming standard Weyl approximations.
• To provide a semiclassical interpretation of Anderson localization by mapping wave interference patterns into a potential landscape with wells and barriers.
• To enable practical applications such as modeling carrier distributions in disordered semiconductors and probing the mobility edge in higher dimensions.

Proposed method

• Define the localization landscape $ u $ as the solution to the Dirichlet problem $ (-\frac{\hbar^2}{2m}\Delta + V)u = 1 $, with appropriate boundary conditions.
• Introduce the auxiliary function $ \varphi = \psi / u $, transforming the original Schrödinger equation into a new equation involving $ W = 1/u $: $ -\frac{\hbar^2}{2m}\left[\frac{1}{u^2}\text{div}(u^2\nabla\varphi)\right] + W\varphi = E\varphi $.
• Establish that $ W $ acts as an effective confining potential, with its maxima (crests) corresponding to the barriers of the localization subregions.
• Use the Agmon metric to describe the exponential decay of $ \psi $, showing that decay occurs predominantly across the barriers of $ W $, not uniformly.
• Apply Weyl’s law to the effective potential $ W $, computing the approximation $ N_W(E) $ of the integrated density of states and comparing it to the true $ N(E) $.
• Validate the method numerically on one-dimensional systems with random (uniform, boolean) and periodic potentials, showing $ N_W(E) $ closely matches $ N(E) $, especially near the mobility edge.

Experimental results

Research questions

• RQ1 How can the long-range exponential decay of Anderson-localized states be explained in terms of a well-defined confining potential?
• RQ2 What is the physical and mathematical role of the function $ W = 1/u $, where $ u $ is the localization landscape?
• RQ3 Can Weyl’s law applied to $ W $ yield a more accurate approximation of the integrated density of states than standard Weyl approximations in disordered systems?
• RQ4 How does the effective potential $ W $ capture the interference patterns responsible for localization in a semiclassical framework?
• RQ5 To what extent does the effective potential $ W $ reveal the structure of localization subregions and their coupling via tunneling?

Key findings

• The function $ W = 1/u $, derived from the localization landscape $ u $, acts as an effective confining potential that determines the boundaries of localization subregions.
• The exponential decay of localized quantum states is not uniform but occurs predominantly across the barriers of $ W $, corresponding to regions of high effective potential.
• The Agmon estimates based on $ W $ accurately reproduce the decay of $ \psi $ down to amplitudes below $ 10^{-7} $, matching numerical results in logarithmic scale.
• The Weyl approximation based on $ W $, denoted $ N_W(E) $, shows a remarkable agreement with the true integrated density of states $ N(E) $ across various 1D systems, including random and periodic potentials.
• In contrast, the standard Weyl approximation using the original potential $ V $ fails to capture the correct density of states, especially near the mobility edge and the lower edge of the conduction band.
• The effective potential $ W $ successfully captures the shift in the lower edge of the conduction band due to disorder or periodicity, reflecting the localization gap observed in Anderson localization.