Bloch-like waves in random-walk potentials based on supersymmetry

TL;DR

This paper introduces a novel method using supersymmetry to create and tune bandgaps in highly disordered random-walk potentials, challenging the notion that order is necessary for bandgap formation.

Contribution

It presents a deterministic approach to generate bandgaps in disordered media by applying supersymmetry, transforming order into disorder while preserving spectral properties.

Findings

Bandgaps can be formed in highly disordered potentials similar to Brownian motion.

The method allows tuning correlations without losing bandgaps.

It creates a family of potentials with Bloch-like eigenstates.

Abstract

Bloch's theorem was a major milestone that established the principle of bandgaps in crystals. Although it was once believed that bandgaps could form only under conditions of periodicity and long-range correlations for Bloch's theorem, this restriction was disproven by the discoveries of amorphous media and quasicrystals. While network and liquid models have been suggested for the interpretation of Bloch-like waves in disordered media, these approaches based on searching for random networks with bandgaps have failed in the deterministic creation of bandgaps. Here, we reveal a deterministic pathway to bandgaps in random-walk potentials by applying the notion of supersymmetry to the wave equation. Inspired by isospectrality, we follow a methodology in contrast to previous methods: we transform order into disorder while preserving bandgaps. Our approach enables the formation of bandgaps in extremely-disordered potentials analogous to Brownian motion, and also allows the tuning of correlations while maintaining identical bandgaps, thereby creating a family of potentials with 'Bloch-like eigenstates'.