[Paper Review] Corner states in second-order mechanical topological insulator

Chun‐Wei Chen, Rajesh Chaunsali+3 more|arXiv (Cornell University)|Sep 8, 2020

Topological Materials and Phenomena36 references4 citations

TL;DR

This paper proposes a continuous hexagonal bolted plate as a second-order mechanical topological insulator, where tuning the bolt radius induces topologically trivial and non-trivial phases. By creating a Z-shaped domain wall between these phases, it experimentally demonstrates one-way corner localization of flexural waves, with highly tunable, non-topological corner states showing superior localization over topological ones.

ABSTRACT

We numerically and experimentally study corner states in a continuous elastic plate with em-bedded bolts in a hexagonal pattern. While preserving C6 crystalline symmetry, the system can transition from a topologically trivial to a non-trivial configuration. We create interfacial corners of 60° and 120° by adjoining trivial and non-trivial topological configurations. Due to the rich interaction between the bolts and the continuous elastic plate, we find a variety of corner states with and without topological origin. Notably, some of the corner states are highly localized and tunable. By taking advantage of this property, we experimentally demonstrate one-way corner localization in a Z-shaped domain wall.

Motivation & Objective

• To design a continuous mechanical platform for second-order topological insulators using a hexagonal array of bolts in an elastic plate.
• To demonstrate topological phase transitions in a mechanical system by tuning the bolt radius to induce trivial and non-trivial band gaps.
• To experimentally verify asymmetric wave localization via a Z-shaped domain wall between topologically distinct regions.
• To explore and exploit non-topological corner states with enhanced localization and tunability for energy flow control.
• To provide a practical, scalable mechanical analog of higher-order topological insulators beyond discrete lattices.

Proposed method

• Using finite element analysis (FEA) to simulate the band structure of a bolted aluminum plate with varying bolt radii (R), where R = a/3 corresponds to a honeycomb lattice with a double Dirac cone at the Γ point.
• Applying a lumped-mass model to analytically determine topological invariants based on rotational symmetry of eigenmodes at high-symmetry points in the Brillouin zone.
• Creating interfacial corners (60° and 120°) by joining regions with R = 0.8a/3 (trivial) and R = 1.1a/3 (non-trivial), forming a Z-shaped domain wall.
• Employing piezoelectric actuators and laser Doppler vibrometry to excite and measure flexural wave responses at specific frequencies.
• Using chirp signals (2–40 kHz) and point-by-point scanning to reconstruct steady-state wave-field patterns and confirm corner state localization.
• Comparing simulated and experimental eigenmode profiles to validate the existence and localization of corner states at specific frequencies (e.g., f = 8.52 kHz).

Experimental results

Research questions

• RQ1 Can a continuous elastic plate with hexagonally arranged bolts support in-gap corner states in a second-order topological insulator phase?
• RQ2 How does tuning the bolt radius (R) affect the topological phase transition from trivial to non-trivial band gap?
• RQ3 What is the origin and localization behavior of corner states formed at 60° and 120° corners between topologically distinct domains?
• RQ4 Can non-topological corner states exhibit stronger localization and tunability than topological ones in such mechanical systems?
• RQ5 Can one-way corner localization be experimentally demonstrated using a Z-shaped domain wall between topologically distinct regions?

Key findings

• The system exhibits a topological phase transition at R = a/3, where a double Dirac cone at the Γ point opens into a band gap upon deviation from this radius.
• At R = 0.8a/3, a trivial band gap emerges (grey rectangle in Fig. 1e), while at R = 1.1a/3, a non-trivial band gap forms, confirmed by eigenmode symmetry analysis.
• Corner states at 60° and 120° corners are observed in simulations and experiments, with the high-frequency state (f = 8.52 kHz) being non-topological and highly localized.
• Experimental measurements at f = 8.49 kHz confirm strong energy confinement only at corner (II), which corresponds to the non-topological corner state, while corner (I) shows no such localization.
• The frequency of the corner state in experiment (8.49 kHz) matches the simulation (8.52 kHz), indicating excellent agreement between theory and measurement.
• One-way wave localization is demonstrated: excitation at the middle of the Z-shaped interface leads to energy propagation and localization only at corner (II), not at corner (I), despite equal distance.