Hierarchical Structures of Quantum Geometric Spectrum in Quasicrystals: A Renormalization-Group Study

Quantum geometry, characterized by the quantum metric and Berry curvature, is a powerful framework for understanding diverse physical phenomena in quantum materials, but its behavior in non-periodic systems remains largely uncharted. Here, we uncover a universal mechanism for the divergent enhancement of the quantum metric in one-dimensional quasiperiodic systems, governed by the interplay of wavefunction criticality and spectral fractality. Using the paradigmatic Fibonacci chain, we demonstrate that the quantum metric displays a hierarchical scaling structure that mirrors the fractal organization of the energy spectrum. A real-space renormalization-group analysis yields an analytic power-law scaling, $\mathcal{G} \propto (\Delta E)^{-k}$, between the quantum metric $\mathcal{G}$ and spectral gap $\Delta E$, with the exponent $k$ dictated by the system's self-similarity. This scaling persists in the critical Aubry-Andr\'e-Harper model but disappears in both its localized and extended phases, confirming its universality across different quasiperiodic paradigms and its unique link to criticality. Our results show that the quantum metric provides a sensitive geometric indicator of quasiperiodic criticality, and highlight quasicrystals as promising platforms for realizing unconventional giant quantum geometric effects beyond the limits of periodic crystals.