Hierarchical Topological States without Dimension Reduction

TL;DR

This paper introduces a novel mechanism to create hierarchical topological states in classical wave systems without reducing dimensions, by controlling domain wall positions in SSH models to generate multiple levels of protected modes.

Contribution

It presents a new approach to generate higher-order topological states through domain wall repositioning, avoiding symmetry breaking or dimension reduction, thus expanding topological classification.

Findings

Hierarchical topological states can be constructed using SSH models.

Controlled domain wall repositioning enables multiple topological levels.

The method preserves symmetry and does not require 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.