Wave functions in the critical phase: A planar Sierpiński fractal lattice

Electronic states play a crucial role in many quantum systems of moir\'e superlattices, quasicrystals, and fractals. As recently reported for Sierpi {n} \'{n}{}ski lattices [Yao et al., 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 Sierpi {n} \'{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 the ``seed lattice'' of the generator and slightly modulated by the dilation pattern and the geometrical self-similarity level. These electronic states are made multifractal by scaling the qth-order inverse participation ratio or fractal dimension, which correlates with the subdiffusion behavior. In the gene pattern, an average state-based multifractal dimension of second order would increase as its Hausdorff dimension increases. Our findings could potentially contribute to understanding quantum transports and single-particle quantum dynamics in fractals.