Quantum Metric Enhancement and Hierarchical Scaling in One-Dimensional Quasiperiodic Systems

TL;DR

This study reveals that one-dimensional quasiperiodic systems exhibit enhanced quantum metrics due to critical wavefunctions and spectral fractality, with potential for novel quantum geometric applications.

Contribution

It uncovers how quasiperiodicity enhances quantum geometry and links spectral fractality with quantum metric properties, expanding understanding beyond periodic crystals.

Findings

Quantum metric is significantly enhanced in quasiperiodic systems.

Quantum metric sharply probes localization transitions in the Aubry-André-Harper model.

Hierarchical spectral structure causes anomalous quantum metric enhancement in Fibonacci chains.

Abstract

The quantum metric, a key component of quantum geometry, plays a central role in a wide range of physical phenomena and has been extensively studied in periodic crystals and moir\'{e} materials. Here, we systematically investigate quantum geometry in one-dimensional (1D) quasiperiodic systems and uncover novel properties that fundamentally distinguish them from both periodic crystals and disordered media. Our comparative analysis reveals that quasiperiodicity significantly enhances the quantum metric -- despite the absence of translational symmetry -- due to the presence of critical wavefunctions with long-range spatial correlations. In the Aubry-Andr\'{e}-Harper model, we show that the quantum metric serves as a sensitive probe of localization transitions, exhibiting sharp changes at the critical point and distinct behaviors near mobility edges. In the Fibonacci chain, characterized by a singular continuous spectrum, we discover an anomalous enhancement of the quantum metric when the Fermi level lies within the minimal gaps of the fractal energy spectrum. Using a perturbative renormalization group framework, we trace this enhancement to the hierarchical structure of the spectrum and the bonding-antibonding nature of critical states across narrow gaps. Our findings establish a fundamental connection between wavefunction criticality, spectral fractality, and quantum geometry, suggesting quasiperiodic systems as promising platforms for engineering enhanced quantum geometric properties beyond conventional crystalline paradigms.