Higher-order topological phases on fractal lattices

TL;DR

This paper predicts the existence of higher-order topological phases, including Majorana modes, on fractal lattices like Sierpiński carpets, expanding topological matter understanding into fractional dimensions.

Contribution

It introduces the concept of higher-order topological phases on quantum fractals, demonstrating realizations of second-order topological insulators and superconductors in fractal geometries.

Findings

Higher-order topological phases can exist on fractal lattices.

Majorana corner modes are supported in fractal topological phases.

Experimental platforms like photonic and acoustic lattices can realize these phases.

Abstract

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 remains vastly unexplored. Here, we theoretically show that a special class of crystalline phases, 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, supporting charged and neutral Majorana corner modes, on planar Sierpi\'{n}ski carpet and triangle fractals, respectively. 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.