What kind of topological states can be found in fractals?*

The periodic table of topological insulators and superconductors classifies various topological phases by identifying their underlying symmetries and the spatial dimension d, where d is an integer. All topological phases show some form of bulk-boundary correspondence, where a nontrivially gapped bulk leads to robust edge states on the boundary. Can a topological phase exist in a nonintegral spatial dimension? What is "bulk-boundary correspondence" in a system where the bulk and boundary are not immediately identifiable? Motivated by these questions we construct topologically nontrivial Hamiltonian on Sierpinksi gasket, a fractal, which has a nonintegral Hausdorff dimension. Restricting to “translationally invariant” deterministic fractals, where each site is equivalently coordinated as any other, we find that such systems are always metals. Interestingly, such a metal is chiral in nature and has two terminal conductance close to unit value, even while the topological index is zero. We also demonstrate “surface states” in yet another fractal, the Sierpinksi tetrahedron, which interestingly has surfaces made of fractals. We describe and discuss various aspects of this physics.