Localization Properties of a Disordered Helical Chain

TL;DR

This work investigates localization in a 1D quasiperiodic chain with a helical geometry that introduces a single long-range $N$th-neighbor hopping. Using exact diagonalization, the authors compute the inverse participation ratio $IPR$, the normalized participation ratio $NPR$, and a composite metric $\eta$ to construct phase diagrams in the $\Delta$-$J_N$ plane. They identify three regimes—fully extended, fully localized, and mixed with mobility edges—and show that the phase structure is governed by the interplay between $N$ and the incommensurate frequency $\beta$, notably producing a near-AA transition when $N$ is Fibonacci. The results provide an arithmetic means to engineer localization in helical lattices, with potential applications in photonics and cold-atom platforms and avenues for further study in finite-size scaling, transport, and many-body effects.

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-André model behavior.

Figures (3)

• Figure 1: Schematic representation of helical tight-binding model. (a) One-dimensional chain representation of the long-range hopping model. Circles illustrate lattice sites, solid lines represent nearest neighbor hopping, and dotted arches show long-range hopping that connects a site to the one located in the next helical winding ($N$th neighbor, here $N=5$). (b) Geometric picture of the same model in an unwrapped helix. Sites are arranged with $5$ lattice points per turn.
• Figure 2: Phase diagrams of the exponent $\eta$ in $\Delta$ and $J_N$ plane for different number of helical turns. (a) $N=40$ (b) $N=200$ (c) $N=377$. Green color denotes a completely localized or extended regime, whereas orange color indicates a mixed regime where localized and extended regimes coexist. The black dots mark the estimated phase boundaries.
• Figure 3: ($J_N=0.5$): Spectrum averaged IPR and NPR versus quasi-periodic potential strength $\Delta$ for helical turns (a) $N=40$, (b) $N=200$, (c) $N=377$. Solid curves shows $\langle\rm IPR\rangle$ whereas dashed curves shows $\langle\rm NPR\rangle$. Gray vertical bands identify the mixed regions.