  Three-dimensional topological insulators support gapless Dirac fermion
surface states whose rich topological properties result from the interplay of
symmetries and dimensionality. Their topological properties have been
extensively studied in systems of integer spatial dimension but the prospect of
these surface electrons arranging into structures of non-integer dimension like
fractals remains unexplored. In this work, we investigate a new class of states
arising from the coupling of surface Dirac fermions to a time-reversal
symmetric fractal potential, which breaks translation symmetry while retaining
self-similarity. Employing large-scale exact diagonalization, scaling analysis
of the inverse participation ratio, and the box-counting method, we establish
the onset of self-similar Dirac fermions with fractal dimension for a
symmetry-preserving surface potential with the geometry of a Sierpinski carpet
fractal with fractal dimension $D \approx 1.89$. Dirac fractal surface states
open a fruitful avenue to explore exotic regimes of transport and quantum
information storage in topological systems with fractal dimensionality.
