Double Dirac cones and topologically nontrivial phonons for continuous square symmetric <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>C</mml:mi><mml:mrow><mml:mn>4</mml:mn><mml:mo>(</mml:mo><mml:mi>v</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:msub></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>C</mml:mi><mml:mrow><mml:mn>2</mml:mn><mml:mo>(</mml:mo><mml:mi>v</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:msub></mml:math> unit cells
Yan Lu, Harold S. Park
Abstract
Because phononic topological insulators have primarily been studied in discrete, graphenelike structures with ${C}_{6(v)}$ or ${C}_{3(v)}$ hexagonal symmetry, an open question is how to systematically achieve double Dirac cones and topologically nontrivial structures using continuous, nonhexagonal unit cells. Here, we address this challenge by presenting a computational methodology for the inverse design of continuous two-dimensional square phononic metamaterials exhibiting ${C}_{4(v)}$ and ${C}_{2(v)}$ 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.