Wave localization in number-theoretic landscapes

TL;DR

This paper explores wave localization in aperiodic structures based on number-theoretic functions, revealing broadband localization and multifractal spectral properties with potential applications in quantum and classical devices.

Contribution

It introduces a new class of deterministic aperiodic systems using number-theoretic functions, demonstrating strong wave localization and multifractal spectral features.

Findings

Eigenmodes are strongly localized across the entire spectrum.

Spectral analysis shows multifractal scaling properties.

Absence of level repulsion indicates broadband localization.

Abstract

We investigate the localization of waves in aperiodic structures that manifest the characteristic multiscale complexity of certain arithmetic functions with a central role in number theory. In particular, we study the eigenspectra and wave localization properties of tight-binding Schr\"{o}dinger equation models with on-site potentials distributed according to the Liouville function $\lambda(n)$, the M\"{o}bius function $\mu(n)$, and the Legendre sequence of quadratic residues modulo a prime (QRs). We employ Multifractal Detrended Fluctuation Analysis (MDFA) and establish the multifractal scaling properties of the energy spectra in these systems. Moreover, by systematically analyzing the spatial eigenmodes and their level spacing distributions, we show the absence of level repulsion with broadband localization across the entire energy spectra. Our study introduces deterministic aperiodic systems whose eigenmodes are all strongly localized in realistic finite one-dimensional systems and provides opportunities for novel quantum and classical devices of particular importance to cold-atom experiments in engineered speckle potentials and enhanced light-matter interactions.