Topological phases on fractal Lattices: Constructions and bulk-boundary correspondence*

Topological crystals, described by universal massive Dirac-Bloch Hamiltonian, feature robust gapless modes at their interfaces (edge, surface, hinges and corners), encoding nontrivial geometry of bulk electronic wavefunctions. However, nature also fosters non-crystalline (a) amorphous materials with no symmetry at all, (b) quasicrystals with only rotational symmetry, and (c) fractals displaying unique self-similarity symmetry. It is, therefore, of fundamental importance to investigate the possibility of realizing topological phases on such material platforms, where the Bloch’s theorem does not apply, due their recent realizations on designer quantum materials and classical metamaterials, such as the topolectric circuits and mechanical lattices.

 

First, I will establish a symmetry-based general principle of constructing real space universal massive Dirac Hamiltonian for topological phases, applicable to any non-crystalline material. This construction will be exemplified for two- and three-dimensional conventional or first-order and higher-order topological systems, with the former ones supporting gapless modes on lower-dimensional boundaries, such as corners and hinges.

 

Next, I will show successful applications of this general approach, specifically higher-order topological insulators, on Sierpinski carpet and triangle fractals. I will show that both systems support zero-energy modes at their outer corners and finite-energy states near the inner corners, which have recently been observed on mechanical lattices. As a penultimate topic, I will show that suitable local inter-orbital pairings give rise to first-order and second-order topological superconductivity on fractal lattices, respectively accommodating Majorana edge and corners modes. In all these cases, appearances of the boundary modes are guaranteed by appropriate real space bulk topological invariant, such as the Bott index and quadrupole moment.

Finally, I will generalize these constructions for non-Hermitian topological fractals, which with periodic boundary conditions exclusively display a novel phenomenon, inner skin effects at their inner boundaries. If time permits, I will also show realizations of topological phases on three-dimensional fractal lattices.