Energy-level statistics in planar fractal tight-binding models

Abstract

In this study, we examine the statistics of level spectra in a noninteracting electron gas confined to a Sierpi\ifmmode \acute{n}\else \'{n}\fi{}ski carpet lattice. These lattices are constructed using two types of the $\mathit{self}$ and $gene$ patterns, and they are categorized by the area-perimeter scaling law. The singularly continuous spectra, along with the nearest level spacing distribution and gap-ratio distribution, reveal a critical phase different from both extended and localized phases. This critical phase differs from the behavior near the Anderson model's metal-insulator transition. The Wigner-like conjecture is confirmed for both lattice classes, indicating Gaussian orthogonal symmetry. A similar observation was made in a quasiperiodic lattice [Phys. Rev. Lett. 80, 3996 (1998)]. The self-similar nature of fractals also leads to level clustering behavior.