Dirac Fractals on the Surface of Topological Insulators

Dimensionality plays a fundamental role in the classification of phases of matter. A canonical example is the surface of 3D topological insulators (TIs) that hosts symmetry-protected 2D Dirac fermions. In this talk, we will discuss a different class of topological surface states that results from the interplay of surface Dirac fermions and fractal geometry. Specifically, we study the consequences of coupling Dirac fermions to a time-reversal symmetric Sierpinski fractal deposited on the surface of a 3D TI with the goal of identifying topological surfaces hosting quantum states with fractal dimension. Employing a numerical analysis of the local density of states and an effective theory, we present evidence for the emergence of the Dirac fractals of non-integer Hausdorff dimension on the surface of TIs. This novel set of Dirac fractals opens a fruitful venue to explore the fate of the bulk-boundary correspondence in TIs and other topological phases when the surface state dimensionality is non-trivial.