Level-spectra Statistics in Planar Fractal Tight-Binding Models

TL;DR

This paper investigates the spectral statistics of noninteracting electrons in Sierpiński Carpet lattices, revealing critical phase behavior and Gaussian orthogonal ensemble characteristics due to fractal topology.

Contribution

It introduces a detailed analysis of level-spectra statistics in fractal lattices, confirming critical phase behavior and Wigner-like distribution in Sierpiński Carpet models.

Findings

Both lattice patterns exhibit critical phase with broken translation symmetry.

Level statistics follow Gaussian orthogonal ensemble predictions.

Fractal topology induces level clustering and critical spectral features.

Abstract

In this communication, we study the level-spectra statistics when a noninteracting electron gas is confined in \textit{Sierpi\'{n}ski Carpet} (\textit{SC}) lattices. These \textit{SC} lattices are constructed under two representative patterns of the $self$ and $gene$ patterns, and classified into two subclass lattices by the area-perimeter scaling law. By the singularly continuous spectra and critical traits using two level-statistic tools\iffalse the nearest spacing distribution and alternative gap-ratio distribution\fi, we ascertain that both obey the critical phase due to broken translation symmetry and the long-range order of scaling symmetry. The Wigner-like conjecture is confirmed numerically since both belong to the Gaussian orthogonal ensemble. An analogy was observed in a quasiperiodic lattice~\cite{Zhong1998Level}. In addition, this critical phase isolates the crucial behavior near the metal-insulator transition edge in Anderson model. The lattice topology of the self-similarity feature can induce level clustering behavior.