Level-spectra Statistics in Planar Fractal Tight-Binding Models
Abstract
In this communication, we study the level-spectra statistics when a noninteracting electron gas is confined in Sierpi\'nski Carpet (SC) lattices. These 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 the nearest spacing distribution and alternative gap-ratio distribution, 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~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.
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