Non-Abelian mechanics of elastic waves in Kagome metamaterials with internal microstructures

Abstract

In this work, the duality of mechanical Kagome lattices with internal microstructures is studied and its non-Abelian characteristics of wave packet evolution are discussed. Based on Bloch's theorem, the dynamical and duality matrices are obtained during elastic wave propagation. By regulating the twisting angles of the Kagome lattices and internal microstructures, global twofold degeneracy and unequal Berry phase can be obtained. A fourfold degeneracy appears at Γ point when the mechanical metamaterial is subjected to a special twisting state. Numerical calculations of non-Abelian mechanics and duality are performed, in which band structures with special twisting angles are the same. In addition, the solution of the Wilson operator is derived by the elastic wave equation and gauge transformation, which has non-exchangeable product and leads to the non-Abelian Berry curvature. If the time-reversal symmetry is broken, non-commutative elastic wave responses can be generated. The generalized polarization and wave propagation become different when the acting sequences of external forces are exchanged.