First-Principles Study And Model Analysis Of Topological Phonon Nodal Chain Materials

Recent studies have shown that some fermions exist in solid systems are similar to those in high energy physics,such as Dirac fermions and Weyl fermions.Further studies have found that the low-energy-excited fermions exist in solids do not exist in high energy physics.A typical example among them is the topological nodal chain,the low energy excitation in the momentum space.The study of topological nodal chain began in the electron system of crystals,and with the deepening of the research,researchers expanded the scope of study to other particle systems such as phonon system.However,topological nodal chain in phonon system has only been realized in a few kinds of artificial acoustic systems,while it has not been reported in atomic crystal（solid）system yet.In addition,whether they exist in electron system or phonon system,whether they are protected by symmorphic or nonsymmorphic symmetry,the chain joints of all the nodal chain involved in previous studies have not appeared at high symmetry points.Is it possible for a chain joint to occur at high symmetry points?Aiming at the shortcomings of these studies,the following work is carried out in this paper:（1）From the perspective of crystal structure symmetry,the rules that must be satisfied when three types of topological phonon nodal chains（Outer type,Inner type and Crossing type）may exist are summarized.For Outer type,it needs to meet the following conditions:there are two intersecting mirror planes with nodal rings;only one band crossing or double degenerate point must exist on the mirror intersection lines;the intersection lines should not be C_n（n=3,4,6）or S_n（n=2,3,6）rotation axis;the intersection lines must pass through the center of the nodal ring.For Inner type and Crossing type,it needs to meet the following conditions:at least two intersecting mirror planes with nodal rings;at least two band intersections or two double degenerate points on the mirror intersection lines;the intersection lines must pass through the center of the nodal ring.（2）Among the above four materials,ideal Outer type topological phonon nodal chain exist in K₂O.The intersections of the two high-frequency optical branches in the phonon spectrum are located at the high symmetry pathΓ-K and the high symmetry point W respectively.The frequency eigenvalues of intersecting bands are opposite near each intersection,indicating that these intersections are protected by the mirror symmetry.Through symmetry analysis and band gap analysis,we find that these bands cross to form nodal chains.We further analyze the formation mechanism of nodal rings and chain joints in detail by a k·p model.To describe the global characteristics of the nodal chain,we build a two-band tight-binding model.The solution of tight-binding model shows that the nodal chain is formed by a surface similar to a six-way valve and a set of mutually perpendicular mirrors.Moreover,the drum-head surface states of the nodal chain in K₂O confirm that the nodal rings forming nodal chains are topologically nontrivial.Because K₂O can be synthesized by experiments easily,it can be a good material platform for topological phonon nodal chain study.