Wave functions in the Critical Phase: a Planar \textit{Sierpi\'{n}ski} Fractal Lattice

TL;DR

This paper investigates the nature of electronic wave functions in the critical phase of Sierpiński carpet fractal lattices, revealing multifractal states influenced by self-similarity and their implications for quantum transport.

Contribution

It demonstrates the multifractal nature of electronic states in Sierpiński carpet lattices and how these states are affected by the lattice's self-similarity and seed structure.

Findings

Electronic states are multifractal and influenced by self-similarity.

Spatial overlap exists between critical states in the lattice.

Multifractal dimensions correlate with the Hausdorff dimension and subdiffusion behavior.

Abstract

Electronic states play a crucial role in many quantum systems of moire superlattices, quasicrystals, and fractals. As recently reported in \textit{Sierpi\'{n}ski} lattices [Phys. Rev. B 107, 115424 (2023)], the critical states are revealed by the energy level-correlation spectra, which are caused by the interplay between aperiodicity and determined self-similarity characters. In the case of the \textit{Sierpi\'{n}ski Carpet}, our results further demonstrate that there is some degree of spatial overlap between these electronic states. These states could be strongly affected by its `seed lattice' of the $generator$, and slightly modulated by the dilation pattern and the geometrical self-similarity level. These electronic states are multifractal by scaling the $q$-order inverse participation ratio or fractal dimension, which correlates with the subdiffusion behavior. In the $gene$ pattern, the averaged state-based multifractal dimension of second-order would increase as its \textit{Hausdoff dimension} increases. Our findings could potentially contribute to understanding quantum transports and single-particle quantum dynamics in fractals.