Analytic expressions for topologically protected edge states in Su–Schrieffer–Heeger model

The Su–Schrieffer–Heeger (SSH) model can demonstrate the topological phase transition of spinless fermions in a two-dimensional square lattice. In this work, we investigate the topologically protected edge states (TESs) in 1D SSH ribbons by using the wave mechanics approach. A concise analytic equation is derived to determine the localization length of TESs. Based on the resulted localization length, the explicit analytic expressions are given for the energy dispersions and wave functions of TESs in 1D SSH ribbons. The effect of the size of the 1D SSH ribbon on the topological phase transition and the critical exponent of the localization length of TESs are further demonstrated. It also shows that the degeneracy of TESs is lifted due to the coupling of the two-side TESs. Our results can elucidate the TESs appearing in nontrivial phase, and can also be employed to optimize the structure of the nanoribbon devices.