Wave functions in the Critical Phase: a Planar $\mathit{\text{Sierpi}}\acute{\mathit{\text{n}}}\mathit{\text{ski}}$ Fractal Lattice

Electronic states play a crucial role in many quantum systems of moire superlattices, quasicrystals, and fractals. As recently reported in $\mathit{\text{Sierpi}}\acute{\mathit{\text{n}}}\mathit{\text{ski}}$ 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 $\mathit{\text{Sierpi}}\acute{\mathit{\text{n}}}\mathit{\text{ski}}\mathit{\text{Carpet}}$, 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 $\mathit{\text{Hausdoff}}\mathit{\text{dimension}}$ increases. Our findings could potentially contribute to understanding quantum transports and single-particle quantum dynamics in fractals.