Topological insulators on fractal lattices: A general principle of construction

Fractal lattices with self-similarity symmetry frequently possess discrete symmetries of parent crystals. We formulate three methods to construct the real space Hamiltonian on a fractal lattice from the topological Bloch Hamiltonian on the parent crystal. On crystals, the nontrivial geometry of the underlying electronic wavefunctions and the crystal symmetries stimulates strong and crystalline topological insulators. To demonstrate, we consider a generalized square lattice Chern insulator model, on which we apply these three methods and showcase the appearance of strong and crystalline Chern insulators on the Sierpinski carpet fractal lattice. We also display the bulk boundary correspondence in each method, and demonstrate the stability of these topological phases against weak disorder.