Rigorous Proof For Non-Existence Of Edge States In The Presence Of Long-Range Hopping

Instead of short-range hopping, we study the effect of long-range (LR) terms on edge states in the system with Type I:"...-P-P-P-P-...", where P and - indicate crystal line (CL) and the distance between nearest neighbor (NN) CL, respectively. Based on lattice model Hamiltonian with LR coupling among CLs and all possible neighbor hopping within each CL, when original infinite crystals have reflection symmetry arid each CL has only one electron mode, we analytically demonstrate that no edge states accommodate in such cut crystals with Type I, due to the transfer matrix in the bulk identical to that in the boundary layers. This explains why edge states can arise in Type I with n (n >= 2) electron modes in each CL, resulting from the transfer matrix in the bulk different from that in the boundary layers. Our result is helpful in understanding novel edge states in one- and two-dimensional (2D) superconductors with LR hopping and pairing.