Level-spectra Statistics in Planar Fractal Tight-Binding Models

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.