# Topological Corner Modes in Graphene-Coated Nanowire Waveguide  
[PERSON]  
1 College of Science, Huazhong Agriculture University, Wuhan 430070, China;  
[EMAIL_ADDRESS] (N.L. ); [EMAIL_ADDRESS] (C.D.) [PERSON]  
1 College of Science, Huazhong Agriculture University, Wuhan 430070, China;  
[EMAIL_ADDRESS] (N.L. ); [EMAIL_ADDRESS] (C.D.) [PERSON]  
2 Hubei Key Laboratory of Optical Information and Pattern Recognition, Wuhan Institute of Technology, Wuhan 430205, China; [EMAIL_ADDRESS]  
Shaolin Ke  
modes emerge at the boundary that is one dimension lower than that of the bulk when the bulk topology of the system is non-trivial. Such a picture is the first-order of topological insulators and obeys the general bulk-edge correspondence [23]. The topological insulators based on graphene plasmonics are investigated in various platforms [33; 34; 35; 36; 37; 38; 39]. The Su-Schriefer-Heeger (SSH) model is proposed in binary graphene sheet arrays [36; 37]. The Jackiw-Rebbi states are investigated in graphene sheets by changing the chemical potential of graphene [38]. The infrared topological insulator and one-way edge state are reported in periodically patterned monolayer graphene which exhibits quite a wide band gap and strong interaction in THz spectra [39]. However, the previous study of SPPs in graphene are limited to first-order topological bound modes without studying high-order topology. Higher-order topological insulators are a newly predicted topological phase which follow a different topological bulk-boundary correspondence [40; 41; 42; 43; 44; 45; 46; 47; 48; 49; 50; 51]. The (\(d\)-\(n\))-dimensional mode is found to be localized at the boundary of the \(n\)th-order of a \(d\)-dimensional topological insulator. For instance, a second-order topological insulator in a 2D system has topologically protected zero-dimensional corner states instead of 1D topological edge states, which is the common case in standard 2D systems [44; 45]. The corner states are also reported in various 3D structures which are figured out to have a third-order topology. In addition to edge and surface states, the modes that are located at corners can also remain quite robust, which enriches the study of topological states. The corner modes that are revealed in quadrupole insulators are extremely stable even when the boundary of the structure is tremendously reshaped [48]. Topological corner modes are studied in many different platforms, such as electric circuits [41], microwave circuits [42; 43], acoustic metamaterials [44], photonic crystal [45; 47], plasmon-polaritonic crystals [46], and photonic ring resonators [48]. In this work, we explore the corner modes in a new platform that is constituted of graphene-coated nanowire waveguides. The waveguides are arranged in a breathing kagome lattice, including the triangle and rhombic nanowires. The topological aspect of the system and the related corner modes in this kind of lattice are originally proposed in [49; 50], using tight binding models, and have been very recently experimentally realized in acoustic resonators [44] and laser written waveguide arrays [51]. The existence of corner modes is not caused by quadrupole polarization but as a consequence of nonvanishing Wannier centers [49; 50]. Here, we numerically calculated the effective mode index and the related electric field distributions of corner modes in graphene-coated nanowire arrays. The existence of corner states seriously relay on the relative value of the spatial spacing between the different nanowires. The triangle waveguides have three equivalent corners and the corresponding corner modes are with the degenerate mode index and tend to localize at all corners. In contrast, even the rhombic waveguides have more corners, and there is only a single corner mode located at one corner. We also compare the eigenvalues and mode distributions of the corner and edge modes.