# Topological states on fractal lattices

## TL;DR

This paper explores how topological states behave on fractal lattices, revealing distinct phases with unique spectral and edge properties, and extends these findings to other fractal structures and topological states.

## Contribution

It demonstrates the existence of topological phases on fractal lattices like the Sierpinski gasket and carpet, introducing new physical insights into topological matter in fractal geometries.

## Key findings

- Identified trivial and topological phases on the Sierpinski gasket.
- Discovered gapless chiral edge states in the topological phase.
- Extended the analysis to higher-order topological insulators on fractals.

## Abstract

We investigate the fate of topological states on fractal lattices. Focusing on a spinless chiral p-wave paired superconductor, we find that this model supports two qualitatively distinct phases when defined on a Sierpinski gasket. While the trivial phase is characterized by a self-similar spectrum with infinitely many gaps and extended eigenstates, the novel "topological" phase has a gapless spectrum and hosts chiral states propagating along edges of the graph. Besides employing theoretical probes such as the real-space Chern number, inverse participation ratio, and energy-level statistics in the presence of disorder, we develop a simple physical picture capturing the essential features of the model on the gasket. Extending this picture to other fractal lattices and topological states, we show that the p+ip state admits a gapped topological phase on the Sierpinski carpet and that a higher-order topological insulator placed on this lattice hosts gapless modes localized on corners.