Double Dirac Cones and Topologically Non-Trivial Phonons for Continuous, Square Symmetric (C$_{4v}$ and C$_{2v}$) Unit Cells

TL;DR

This paper introduces a computational method to design continuous square phononic metamaterials with double Dirac cones and topologically protected edge states, expanding topological phononics beyond hexagonal lattices.

Contribution

It presents a novel inverse design approach for continuous square unit cells with C4v and C2v symmetry to achieve topologically non-trivial phononic structures.

Findings

Double Dirac degeneracy achieved in square phononic metamaterials.

Helical edge states observed at interfaces between topologically distinct structures.

Potential for quantum spin Hall effect-based phononic transport beyond hexagonal lattices.

Abstract

Because phononic topological insulators have primarily been studied in discrete, graphene-like structures with C$_{6}$ or C$_{3}$ hexagonal symmetry, an open question is how to systematically achieve double Dirac cones and topologically non-trivial structures using continuous, non-hexagonal unit cells. Here, we address this challenge by presenting a novel computational methodology for the inverse design of continuous two-dimensional square phononic metamaterials exhibiting C$_{4v}$ and C$_{2v}$ symmetry. This leads to the systematic design of square unit cell topologies exhibiting a double Dirac degeneracy, which enables topologically-protected interface propagation based on the quantum spin Hall effect (QSHE). Numerical simulations prove that helical edge states emerge at the interface between two topologically distinct square phononic metamaterials, which opens the possibility of QSHE-based pseudospin-dependent transport beyond hexagonal lattices.