Higher-Order Topological Phases on Quantum Fractals

Electronic materials harbor a plethora of exotic quantum phases, ranging from unconventional superconductors to non-Fermi liquids, and more recently topological phases of matter. While these quantum phases in integer dimensions are well characterized by now, their presence in fractional dimensions remain vastly unexplored. Here we theoretically show that a special class of crystalline, namely higher-order topological phases that via an extended bulk-boundary correspondence feature robust gapless modes on lower dimensional boundaries, such as corners and hinges, can be found on a representative family of fractional materials, \emph{quantum fractals}. To anchor this general proposal, we demonstrate realizations of second-order topological insulators and superconductors, respectively supporting charged and neutral Majorana corner modes, on planar Siperpenski carpet and triangle fractals. These predictions can be experimentally tested on designer electronic fractal materials, as well as on various highly tunable metamaterial platforms, such as photonic and acoustic lattices.