Edge modes in finite-size systems with different edge terminals

We investigate the behavior of edge modes in the presence of different edge terminations and long-range(LR)hopping.Here,we mainly focus on such model crystals with two different types of structures(type I:“…-P-Q-P-Q-…”and type II:“…=P-Q=P-Q=…”),where P and Q represent crystal lines(CLs),while the symbols“-”and“=”denote the distance between the nearest neighbor(NN)CLs.Based on the lattice model Hamiltonian with LR hopping,the existence of edge modes is determined analytically by using the transfer matrix method(TMM)when different edge terminals are taken into consideration.Our findings are consistent with the numerical results obtained by the exact diagonalization method.We also notice that edge modes can exhibit different behaviors under different edge terminals.Our result is helpful in solving novel edge modes in honeycomb crystalline graphene and transition metal dichalcogenides with different edge terminals.