Critical quantum states and hierarchical spectral statistics in a Cantor potential

TL;DR

This paper investigates how a fractal Cantor potential influences quantum spectral properties, revealing hierarchical spectral structures, critical eigenstates, and the impact of fractal geometry on quantum criticality.

Contribution

It demonstrates the direct link between fractal geometry and hierarchical spectral statistics, highlighting critical quantum states in a Cantor potential.

Findings

Hierarchical, filamentary level-spacing structure observed.

Eigenstates exhibit multifractal, critical behavior.

Spectral scaling transitions from fractal to semiclassical regimes.

Abstract

We study the spectral statistics and wave-function properties of a one-dimensional quantum system subject to a Cantor-type fractal potential. By analyzing the nearest-neighbor level spacings, inverse participation ratio (IPR), and the scaling behavior of the integrated density of states (IDS), we demonstrate how the self-similar geometry of the potential is imprinted on the quantum spectrum. The energy-resolved level spacings form a hierarchical, filamentary structure, in sharp contrast to those of periodic and random systems. The normalized level-spacing distribution exhibits a bimodal structure, reflecting the deterministic recurrence of spectral gaps. A multifractal analysis of eigenstates reveals critical behavior: the generalized fractal dimensions $D_q$ lie strictly between the limits of extended and localized states, exhibiting a distinct $q$-dependence. Consistently, the IPR indicates the coexistence of quasi-extended and localized features, characteristic of critical wave functions. The IDS shows anomalous power-law scaling at low energies, with an exponent close to the Hausdorff dimension of the underlying Cantor set, indicating that the geometric fractality governs the spectral dimensionality. At higher energies, this scaling crosses over to the semiclassical Weyl law. Our results establish a direct connection between deterministic fractal geometry, hierarchical spectral statistics, and quantum criticality.