Transport fingerprints of helical edge states in Sierpiński tapestries

Recently, synthesis and experimental research of fractalized materials has evolved in a paradigmatic crossroad with topological states of matter. Here, we present a theoretical investigation of the helical edge transport in Sierpiński carpets (SCs), combining the Bernevig–Hughes–Zhang model with the Landauer transport framework. By starting from a pristine two-dimensional topological insulator, the results reveal vanishing and reentrant resonant transport modes enabled for increased SC fractal generation. We observe that fractal with superior hierarchy inherits characteristics due to self-similarity and present conductance patterns resembling a miniband transport picture with fractal fingerprints. Real-space mapping of emerging resonant and antiresonant states provides an unprecedented view of helical-edge currents encoded in these intricate geometries and their multiple edges, underscoring the significance and consistency of our findings.