Topological insulators are materials that are insulating in the bulk within the bandgap, yet exhibit conductive states on their surface at frequencies within that same bandgap. These surface states are topologically protected, meaning they cannot be easily disrupted by local perturbations. In general, a material of n‑dimensions can host n‑1-dimensional topological boundary states. If the symmetry protecting these states is further broken, a bandgap can open between the n-1-dimensional states, enabling the emergence of n-2-dimensional topological states. For example, a 3D material can host 2D protected surface states, and breaking additional symmetry can create a bandgap between these surface states, allowing for protected 1D edge states. A material undergoing such a process is said to exhibit a phenomenon known as a higher-order topological insulator. In general, higher-order topological states appear in dimensions one lower than the parent topological phase due to the further unit-cell symmetry reduction. This requires at least a 2D lattice for second-order states, with the maximal order in 3D systems being three.

The researchers here introduce a new method for repeatedly opening the bandgap between topological states and generating new states within those gaps in an unbounded manner – without breaking symmetries or reducing dimensions. Their approach creates hierarchical topological insulators by repositioning domain walls between different topological regions. This process opens bandgaps between original topological states while preserving symmetry, enabling the formation of new hierarchical states within the gaps. Using one‑ and two‑dimensional Su–Schrieffer–Heeger models, they show that this procedure can be repeated to generate multiple, even infinite, hierarchical levels of topological states, exhibiting fractal-like behavior reminiscent of a Matryoshka doll. These higher-level states are characterized by a generalized winding number that extends conventional topological classification and maintains bulk-edge correspondence across hierarchies.

The researchers confirm the existence of second‑ and third-level domain‑wall and edge states and demonstrate that these states remain robust against perturbations. Their approach is scalable to higher dimensions and applicable not only to quantum systems but also to classical waves such as phononics. This broadens the definition of topological insulators and provides a flexible way to design complex networks of protected states. Such networks could enable advances in electronics, photonics, and phonon‑based quantum information processing, as well as engineered structures for vibration control. The ability to design complex, robust, and tunable hierarchical topological states could lead to new types of waveguides, sensors, and quantum devices that are more fault-tolerant and programmable.

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Interacting topological insulators: a review by Stephan Rachel (2018)