Localization Transitions in a Half-Filled Helical Aubry-Andr\'e Model

TL;DR

This study investigates localization transitions in a quasiperiodic lattice with long-range hopping, revealing how helical tunneling influences the critical potential and phase stability using polarization-based diagnostics.

Contribution

It introduces a method to analyze localization in a helical Aubry-André model with Nth-neighbor hopping, extending the understanding of phase transitions in quasiperiodic systems.

Findings

Stronger helical hopping stabilizes the extended phase.

Critical potential shifts upward with increased helical hopping.

Pronounced spikes in N-dependence due to commensurability effects.

Abstract

We study localization in a one-dimensional quasiperiodic lattice obtained by extending the Aubry-Andr\'e model with an additional $N$th-neighbor hopping term of strength $J_{N}$. This long-range tunneling couples successive windings of an effective helical chain and introduces a second control parameter beyond the quasiperiodic potential strength $\Delta$. Working with noninteracting fermions (typically at half filling), we diagnose the delocalization-localization transition using extensions of the modern theory of polarization. Specifically, we compute the polarization amplitudes of the many-body Slater-determinant ground state and construct a geometric Binder cumulant from polarization amplitudes. The critical potential where the localization transition happens is extracted from the sign change (zero crossing) of the geometric Binder cumulant. We map critical potential as a function of $J_N$ and the helical range $N$, finding that stronger helical hopping generally stabilizes the extended phase (shifting critical potential upward), while the $N$-dependence can display pronounced commensurability-induced spikes. We further compare the geometric Binder cumulant with the Fermi gap, which remains near zero at small values of potential and opens in the same parameter regime where the geometric Binder cumulant departs from extended phase. Finally, to take a controlled thermodynamic limit along Fibonacci system sizes, we employ a Zeckendorf-shift construction that fixes the many-body sector consistently as system size goes to infinity.