Hierarchical Topological States without Dimension Reduction

Abstract

Topological insulators exhibit boundary states protected by bulk band topology, a principle first established in quantum systems and later extended to classical waves, including phononics. Conventionally, an $n$-dimensional bulk with nontrivial topology hosts $(n-1)$-dimensional topologically protected boundary states, which may be further gapped out by breaking the symmetry that protects them, potentially leading to the emergence of $(n-2)$-dimensional, or even lower-dimensional topological states, as in higher-order topological insulators. In this work, we introduce an alternative mechanism for gapping out topological states and forming new topological modes within the resulting gap without further unit-cell symmetry breaking or dimension reduction. Using one- and two-dimensional Su-Schrieffer-Heeger (SSH) models, we show that controlled repositioning of topological domain walls enables the construction of hierarchical unit cells that gap out the original domain-wall states while preserving the underlying symmetry. This process produces higher-hierarchical-level topological states, characterized by a generalized winding number, and can be iterated to realize multiple - potentially infinite - hierarchical levels of topological states. Our approach expands the conventional topological classification and offers a versatile route for engineering complex networks of protected modes in higher dimensions.