Statistics of Spectra for One-dimensional Quasi-Periodic Systems at the Metal-Insulator Transition

TL;DR

This paper investigates the spectral statistics of one-dimensional quasi-periodic systems at the metal-insulator transition, revealing distinct statistical behaviors at critical points, lines, and regions, including power-law and exponential distributions.

Contribution

It provides a detailed characterization of spectral statistics at the transition, identifying specific distribution forms and exponents for critical points, lines, and regions, which advances understanding of spectral behavior in these systems.

Findings

Bandwidth distribution near critical points follows power-law or exponential forms.

Level spacing distribution exhibits inverse power-law behavior.

Distinct spectral statistical behaviors are identified at different critical regions.

Abstract

We study spectral statistics of one-dimensional quasi-periodic systems at the metal-insulator transition. Several types of spectral statistics are observed at the critical points, lines, and region. On the critical lines, we find the bandwidth distribution $P_B(w)$ around the origin (in the tail) to have the form of $P_B(w) \sim w^{\alpha}$ ($P_B(w) \sim e^{-\beta w^{\gamma}}$) ($\alpha, \beta, \gamma > 0 $), while in the critical region $P_B(w) \sim w^{-\alpha'}$ ($\alpha' > 0$). We also find the level spacing distribution to follow an inverse power law $P_G(s) \sim s^{- \delta}$ ($\delta > 0$)