By extending the concept of higher-order topological phases to fractal lattices, twisted Moire superlattices, and square-root constructions, we achieve the following progresses:

1. Fractal higher-order topological phases

Observation of fractal higher-order topological states in acoustic metamaterials

https://www.sciencedirect.com/science/article/abs/pii/S2095927322004273

Higher-order topological phase in an acoustic fractal lattice

https://www.sciencedirect.com/science/article/pii/S2095927322004431?via%3Dihub

2. higher-order topological phase in twisted Moire superlattices with ultra-strong interlayer couplings

https://journals.aps.org/prapplied/abstract/10.1103/PhysRevApplied.17.034061

3. higher-order topological phase in square-root systems

https://journals.aps.org/prapplied/abstract/10.1103/PhysRevApplied.19.024023

4. We also measured the fractional topological charges in acoustic higher-order topological insulators

https://journals.aps.org/prapplied/abstract/10.1103/PhysRevApplied.19.034073