Analytical solution of Dirac degeneracy in one-dimensional nonlocal phononic crystals

We present an analytical investigation of Dirac cones in one-dimensional phononic crystals, focusing on the role of beyond-nearest-neighbor (BNN) interactions. Unlike the trivial Dirac cone that appears at the Brillouin zone (BZ) boundary, we systematically introduce nontrivial Dirac cones at any location within the BZ using BNN interactions. We present results for Dirac cone configurations up to the fifth BNN interactions and show the analytical method of Dirac cone solutions for any nonlocal metamaterials. Moreover, we elucidate the dependence of Dirac cone types on stiffness values. Additionally, time-domain simulations at the Dirac degeneracy validate our findings, offering insights into the dynamic behavior of these nontrivial Dirac cones.