Hierarchical topological states without dimension reduction.

Area of Science:

• Condensed Matter Physics
• Topological Materials
• Classical Wave Phenomena

Background:

• Topological insulators feature boundary states protected by bulk topology, a concept originating in quantum mechanics and extended to classical waves like phononics.
• Conventional topological states are (n-1)-dimensional in an n-dimensional bulk and can be gapped out, potentially creating lower-dimensional states (higher-order topological insulators).

Purpose of the Study:

• To introduce an alternative mechanism for gapping out topological states and forming new topological modes.
• To achieve this without breaking unit-cell symmetry or reducing dimensionality.
• To expand the conventional topological classification and enable engineering of protected modes.

Main Methods:

• Utilized one- and two-dimensional Su-Schrieffer-Heeger models.
• Employed controlled repositioning of topological domain walls to construct hierarchical unit cells.
• Analyzed the emergence of higher-hierarchical-level topological states characterized by a generalized winding number.

Main Results:

• Demonstrated a method to gap out existing domain-wall states while preserving underlying symmetry.
• Successfully produced higher-hierarchical-level topological states through iterated domain wall repositioning.
• Showcased the potential for realizing multiple, potentially infinite, hierarchical levels of topological states.

Conclusions:

• The proposed approach offers a versatile route for engineering complex networks of protected topological modes.
• This work expands the conventional topological classification beyond existing frameworks.
• The method provides a new pathway for designing topological states in higher dimensions.