Emergent Quantum Fractality on the Surface of Topological Insulators

Three-dimensional topological insulators (TIs) are bulk insulators with Dirac fermion surface states protected by time-reversal symmetry and charge conservation. The rich topological structure of these states comes from a combination of symmetries and dimensionality. The topological properties of these states 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 talk, I will discuss the states that result from the coupling of surface Dirac fermions with time-reversal symmetric fractal networks deposited on TI surfaces. Theoretical modeling and numerical analysis support the existence of emerging fermionic states with non-integer Hausdorff dimensions. These fractal surface states open a fruitful venue to explore novel regimes of transport and quantum information storage. Furthermore, they can give insights into the fate of the bulk-boundary correspondence in TIs and other topological phases when the surface state dimensionality is non-trivial.