Fractal Surface States in Three-Dimensional Topological Quasicrystals

We study topological states of matter in quasicrystals, which do not rely on crystalline orders. In the absence of a bandstructure description and spin-orbit coupling, we show that a three-dimensional quasicrystal can nevertheless form a topological insulator. It relies on a combination of noncrystallographic rotational symmetry of quasicrystals and electronic orbital space symmetry, which is the quasicrystalline counterpart of topological crystalline insulator. The resulting topological state obeys a non-trivial twisted bulk-boundary correspondence and lacks a good metallic surface. The topological surface states, localized on the top and bottom planes respecting the quasicrystalline symmetry, exhibit a new kind of multifractality with probability density concentrates mostly on high symmetry patches. They form a near-degenerate manifold of 'immobile' states whose number scales proportionally with the macroscopic sample size. This can open the door to a novel platform for topological surface physics distinct from the crystalline counterpart.