  In recent years, synthesis and experimental research of fractalized materials
has evolved in a paradigmatic crossover with topological phases of matter. We
present here a theoretical investigation of the helical edge transport in
Sierpinski carpets (SCs), combining the Bernevig-Hughes-Zhang (BHZ) model and
the Landauer approach. Starting from a pristine two-dimensional topological
insulator (2DTI), according to the BHZ model, our results reveal resonant
transport modes when the SC fractal generation reaches the same scale as the
space discretization; these modes are analyzed within a contour plot mapping of
the local spin-polarized currents, shown spanned and assisted by inner-edge
channels. From such a deeply fractalized SC building block, we introduce a rich
tapestry formed by superior SC hierarchies, enlightening intricate patterns and
unique fingerprints that offer valuable insights into how helical edge
transport occurs in these fractal dimensions.
