Phononic linear and quadratic nodal points in monolayer XH (X=Si, Ge, Sn)

Topological phases in two-dimensional (2D) systems have been attracting tremendous attention since the discovery of graphene. Since the experimental probing could proceed in the whole phonon spectrum, intensive research effort has been devoted to the topological quantum phases in phononic systems. Via first-principles calculations, we predict that a family of 2D hexagonal materials, XH (X = Si, Ge, Sn), hosts ideal linear nodal points (LNPs) and quadratic phononic nodal points (QNPs). Specifically, the LNPs appear at the two inequivalent valleys, akin to the 2D Dirac point in graphene, connecting by an edge arc. The QNP is pinned at the Γ point, two edge states emerge from their projections. Remarkably, both LNPs and QNP enjoy an emergent chiral symmetry, we then show that they feature nontrivial topological charges. As a consequence, our work discusses the nodal points in the phonon spectrum of 2D materials and provides ideal candidates to study the topology for bosonic systems.