Herein, based on first-principles calculations and symmetry analysis, we propose the coexistence of gapped and gapless topological phonon states in Kekulé-order graphene. 48,49 However, the topological phonon states in Kekulé-order graphene have not been touched by other researchers to this date. Note that a recently experimentally prepared 2D material, Kekulé-order graphene, 48 is a fantastic playground for investigating chiral symmetry breaking related physics, flatband instabilities, self-energy dynamics, and the mode-specific phonon threshold effect. Therefore, it can be questioned whether the three-terminal Weyl complex, whose phonon branches consist of three WPs, exists in a 2D material. 42 postulated a symmetry-protected three-terminal Weyl complex with phonon branches that comprise one double WP and two single WPs (a total of three WPs) in 3D realistic materials with trigonal, hexagonal, and cubic lattices. However, extensive research has been conducted on the coexistence of various forms of the nodal points in a 3D single lattice. Our knowledge of the coexistence of different categories of nodal points in two dimensions is still rudimentary. However, only some candidate materials 37–41 with Dirac point phonons, linear Weyl point (LWP) phonons, or quadratic nodal point (QNP) phonons have been proposed based on first-principles calculations and symmetry analysis. Compared to 3D materials, 2D materials with less symmetrical constraints may more intuitively display the clean characteristics of topological phonons. On the other hand, research into gapless topological phonon states in 2D materials is very limited. Meanwhile, gapless and gapped topological phonon states in two-dimensional (2D) materials have received less attention. To date, the gapless topological phonon states of three-dimensional (3D) materials, 11–25 which include nodal point phonons, nodal line phonons, and nodal surface phonons in their phonon curves, have been the focus of most studies on topological phonons. This enables the detection of nontrivial topological phonons across the entire terahertz phonon spectrum and makes phononic systems significantly more flexible than electronic systems. 6–10 Unlike electronic systems, phononic systems have no concept of the Fermi level. Since 2010, findings on the Berry phase and topological physics have been applied to phonons, resulting in the developing field of topological phononics. Phonons play a significant role in the thermal properties of a material, interact closely with other quasiparticles or collective excitations, and are responsible for extraordinary effects (such as superconductivity). Phonons 1–5 are bosonic excitations in crystals they denote the collective vibrations of the atomic lattice. Our work paves the way for new advancements in topological phononics comprising gapless and gapped topological phonons. Our study not only promotes 2D Kekulé-order graphene as a concrete material platform for exploring the intriguing physics of phononic second-order topology but also proposes the coexistence of different categories of Weyl phonons, i.e., a Weyl complex that comprises two LWPs and one QNP, in two dimensions. Moreover, the difference between the phononic Weyl pair and the phononic Weyl complex was investigated in detail. For the gapless topological phonons, 2D Kekulé-order graphene hosts a phononic Weyl pair and a phononic Weyl complex around 7.54 and 47.3 THz (39.2 THz), respectively. For the gapped topological phonons, 2D Kekulé-order graphene hosts phononic real Chern insulator states, i.e., second-order topological states, and corner vibrational modes inside frequency gaps at 27.96 and 37.065 THz. This is the first work to investigate rich gapped and gapless topological phonon states in experimentally feasible 2D materials. In this work, we chose the recently experimentally prepared two-dimensional (2D) Kekulé-order graphene as a target to propose the coexistence of gapless and gapped topological phonon states in its phonon curves. The conceptual framework of topological states has recently been extended to bosonic systems, particularly phononic systems.
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