Ideal topological Weyl complex phonons in two dimensions

The advent of topological phonons has been attracting tremendous attention. However, studies in two-dimensional (2D) systems are limited. Here, we reveal a 2D novel combination of Weyl phonons - a Weyl complex composed of two linear Weyl nodes and one quadratic Weyl node. This Weyl complex consists of crossing points of two specific branches. We show that the coexistence of threefold symmetry - rotation symmetry, inversion symmetry, and time-reversal symmetry - could lead to the presence of the Weyl complex. Based on the symmetry requirement, we further construct the tight-binding model and effective k·p model for characterizing the Weyl complex. Moreover, due to the presence of the spacetime inversion symmetry, the linear and quadratic Weyl nodes feature a quantized (π and 2π) Berry phase, thus defining the corresponding topological charge. Furthermore, Weyl complexes consisting of Weyl points possess an emergent chiral symmetry, an integer topological charge is thus defined. Then, distinguished phenomena for the Weyl complex are studied, in particular, the edge states with three terminals. Our work predicts the presence of this novel 2D topological phase, and provides the symmetry guidance to realize it. Based on the first-principles calculations, we identify an existing material Cu₂Si, as a concrete example to demonstrate the presence of the Weyl complex, and also study the phase transition under symmetry breaking.