# Non-conventional topological band properties and gapless helical edge states in elastic phononic waveguides with Kekul\'e distortion

## TL;DR

This paper demonstrates that elastic phononic waveguides with Kekule9 distortion can host topologically protected gapless helical edge states, revealing novel pseudospin textures and Berry curvature patterns in a classical elastic system.

## Contribution

It introduces a Kekule9-distorted elastic waveguide that exhibits topological properties similar to quantum spin Hall systems, with unique pseudospin textures and edge states.

## Key findings

- Existence of nondegenerate pseudospin states
- Presence of a 6-lobe pseudospin texture and Berry curvature pattern
- Support for gapless helical edge states in elastic phononic systems

## Abstract

This study investigates the topological behavior of a continuous elastic phononic structure characterized by a 3-term Kekul\'e distortion. The elastic waveguide consists of a hexagonal unit cell whose geometric dimensions are intentionally perturbed according to a generalized Kekul\'e scheme. The resulting structure exhibits an effective Hamiltonian that resembles a quantum spin Hall system, hence suggesting that the waveguide can support helical topological edge states. Using first-principles calculations, we show the existence of nondegenerate pseudospin states and a very peculiar 6-lobe pseudospin texture and Berry curvature pattern. Important insights are also provided concerning the topological states in the Kekul\'e lattice, so far considered indistinguishable, and their critical role in enabling unique gapless edge states, typically not achievable in phononic systems.