Localization Properties of a Disordered Helical Chain

TL;DR

This paper investigates how disorder affects electron localization in a one-dimensional helical chain with both nearest-neighbor and long-range tunneling, revealing distinct phases and the influence of Fibonacci numbers on localization.

Contribution

It introduces a phase map for localization in a disordered helical chain considering long-range tunneling and explores the special case of Fibonacci-number sites per turn.

Findings

Identifies three localization regimes: extended, localized, and mixed.

Shows the phase boundary's dependence on the number of sites per turn.

Demonstrates the special case where Fibonacci numbers simplify the phase behavior.

Abstract

We study the localization properties of the quasiperiodic one-dimensional helical chain with two tunneling paths: nearest-neighbor and a long-range hop that connects sites of consecutive helical turns. Using exact diagonalization, we quantify localization employing the inverse participation ratio (IPR) and the normalized participation ratio (NPR), and combine them into a single measure to create a phase map. The resulting diagrams reveal three regimes: a completely extended phase, a completely localized phase, and a mixed domain where localized and extended states coexist. In the diagrams, we investigate the behaviors of tightly and loosely wound helices and examine a special case where the number of sites per turn is a Fibonacci number. For moderate numbers of sites per helical turn, the mixed region is broad and also shifts with the long-range coupling. When the turn size is a Fibonacci number, the phase boundary becomes nearly horizontal and the mixed region fades out, effectively recovering the standard Aubry-Andr\'e model behavior.